H. W. Gould, Articles in Professional Journals

THIS IS AN ABBREVIATED LIST

A detailed listing of over 200 publications and 100 pending is available. {Math. Reviews citations (MR) are noted; many papers have also been reviewed in the German Zentralblatt (ZBL), or in the Russian Referativnii Zhurnal-Matematika (RZ).}

  1. Note on a paper of Grosswald, Amer. Math. Monthly, 65(1954), 251-253.
    MR 15(1954), p.685.
  2. Some generalizations of Vandermonde's convolution, Amer. Math.
    Monthly, 63(1956), 84-91. MR 17(1956), p. 702.
  3. Final analysis of Vandermonde's convolution, Amer. Math. Monthly,
    64(1957), 409-415. MR 19(1958), p.379.
  4. A theorem concerning the Bernstein polynomials, Math. Mag., 31(1958),
    259-264. MR 20(1959), #5997.
  5. Note on a paper of Sparre Andersen, Math. Scand., 6(1958), 226-230.
    MR 21(1960), p.491.
  6. Dixon's series expressed as a convolution, Nordisk Mat. Tidsk., 7(1959),
    73-76. Summary on p. 96. MR 22(1961), #115
  7. Note on a paper of Steinberg, Math. Mag., 33(1959), 46-48.
    MR 21(1960), #4924
  8. An iterative approximation for finding the n-th root of a number,
    Math. Mag., 33(1959), 61-69. MR 23(1962) #B556.
  9. Generalizations of a theorem of Jensen concerning convolutions, Duke
    Math. J., 27(1960), 71-76. MR 22(1961), #5585.
  10. Stirling number representation problems, Proc. Amer. Math. Soc.,
    11(1960), 447-451. MR 22(1961), #5586.
  11. The Lagrange interpolation formula and Stirling numbers, Proc. Amer.
    Math. Soc., 11(1960), 421-425. MR 22(1961), #5588.
  12. A q-binomial coefficient series transform, Bull. Amer. Math. Soc., 66
    (1960), 383-387. MR 22(1961), #9764.
  13. Fjeldstad's series and the q-analog of a series of Dougall, Math. Scand.,
    8(1960), 198-200. MR 23(1962), #A1546.
  14. The q-series generalization of a formula of Sparre Andersen, Math.
    Scand., 9(1961), 90-94. MR 23(1962), #A810.
  15. Note on a Paper of Klamkin concerning Stirling numbers, Amer. Math.
    Monthly, 68(1961), 477-479. MR 23(1962), #A1548.
  16. A series transformation for finding convolution identities, Duke Math. J.,
    28(1961), 193-202. MR 23(1962) #A1216.
  17. The q-Stirling numbers of first and second kinds,. Duke Math. J.,
    28(1961), 281-289. MR 23(1962), #A99.
  18. Some relations involving the finite harmonic series, Math. Mag.,
    34(1961), 317-321. MR 25(1963), #2000.
  19. A generalization of a problem of L. Lorch and L. Moser, Canadian Math.
    Bull., 4(1961), 303-305.
  20. A binomial identity of Greenwood and Gleason, The Mathematics Student,
    29(1961), 53-57. MR 29(1965), #5754.
  21. (with A. T. Hopper) Operational formulas connected with two
    generalizations of Hermite polynomials, Duke Math. J., 29(1962),
    51-63. MR 24(1962), #A2689.
  22. (with Remy Landau), Floating primes, Recreational Math. Magazine,
    Issue No.8, April 1962, 34-35.
  23. Congruences involving sums of binomial coefficients and a formula of
    Jensen, Amer. Math. Monthly, 69(1962), 400-402.
    MR 38(1969), #2033.
  24. (with T. A. Chapman) Some curious limits involving definite integrals,
    Amer. Math. Monthly, 69(1962), 651-653. ZBL 108(1964), 270-
    271.
  25. A new convolution formula and some new orthogonal relations for
    inversion of series, Duke Math. J., 29(1962), 393-404. MR 25(1963),
    #3333.
  26. Generalizations of an integral formula of Bateman, Duke Math. J.,
    29(1962), 475-479. MR 25(1963), #3201.
  27. Theory of binomial sums, Proc. W. Va. Acad. Sci., 34(1962), 158-
    162(1963). MR 36(1968), #3660.
  28. (with William Squire) Maclaurin's second formula and its generalization,
    Amer. Math. Monthly, 70(1963), 44-52. MR 26(1963), #4073.
  29. Operational Recurrences involving Fibonacci numbers, Fibonacci Quart.,
    1(1963), No.1, 30-33. MR 27(1964), #1402.
  30. Note on two binomial coefficient sums found by Riordan, Annals of Math.
    Stat., 34(1963), 333-335. MR 26(1963), #47.
  31. The q-series analogue of a formula of Szily, Scripta Mathematica, 26(1961),
    155-157 (1963). MR 26(1963), #6073.
  32. Generating functions for products of powers of Fibonacci and Lucas
    numbers, Fibonacci Quart., 1(1963) , No.2, 1-16. MR 33(1967), #93.
  33. Generalization of a bracket function formula of L. Moser, Canadian
    Math. Bull. 6(1963), 275-277. ZBL 113(1965), p. 267.
  34. Isotone and antitone fractions, Math. Magazine, 36(1963), 240- 244.
    ZBL 116(1965), p. 41; RZ 1964, #7b3.
  35. A general Túran expression for the Zeta function, Canadian Math. Bull.
    6(1963), 359-366. MR 28(1964), #2099.
  36. Fibonacci exponentials and generalizations of Hermite polynomials,
    Fibonacci Quart., 1(1963), No.4, 31-34. ZBL 147(1968), p. 21.
  37. Sums of logarithms of binomial coefficients, Amer. Math. Monthly,
    71(1964), 55-58. ZBL 123(1966), p. 2. RZ 1964), #10A134.
  38. The functional operator Tf(x) = f(x+a)f(x+b) - f(x)f(x+a+b), Math. Mag.
    37(1964), 38-46. ZBL 123(1968), p. 293. RZ 1964, #12b5.
  39. (with L. Carlitz) Bracket function congruences for binomial coefficients,
    Math. Mag., 37(1964), 91-93. ZBL 128(1967), p. 41.
  40. A new series transform with applications to Bessel, Legendre, and
    Tchebycheff polynomials, Duke Math. J., 31(1964), 325-334.
    MR 28(1964), #4272.
  41. Distance of closest approach of an artificial earth satellite, Proc. W. Va.
    Acad. Sci., 36(1964), 169-174 (1964).
  42. The operator (a^x)Delta)^n and Stirling numbers of the first kind, Amer. Math.
    Monthly, 71(1964), 850-858. MR 29(1965), #4709.
  43. Binomial coefficients, the bracket function, and compositions with
    relatively prime summands, Fibonacci Quart., 2(1964), 241-260.
    MR 30(1965), #1090.  (to Prof.Alfred T. Brauer on his 70th birthday)
  44. The construction of orthogonal and quasi-orthogonal number sets,
    Amer. Math. Monthly, 72(1965), 591-602. MR 31(1966), #2161.
  45. An identity involving Stirling numbers, Annals of Inst. of Statistical
    Math. (Tokyo), 17(1965), 265-269. MR 31(1966), #5808.
  46. Non-Fibonacci numbers, Fibonacci Quart., 3(1965), 177-183.
    MR 35(1968), #118.
  47. A variant of Pascal's triangle, Fibonacci Quart., 3(1965), 257-271.
    Corrections of misprints, ibid., 4(1966), 62. MR 36(1968), #5006.
  48. Inverse series relations and other expansions involving Humbert
    polynomials, Duke Math. J., 32(1965), 697-711. MR 32(1966), #5948.
  49. Cale's rule for planetary distances, Proc. W. Va. Acad. Sci., 37(1965),
    243-257 (1966).
  50. Note on recurrence relations for Stirling numbers, Publ. l'Inst. Math.
    Belgrade, Nouv.Sér. 6(20)(1966), 115-119. MR 34(1967), #4190.
  51. (with Josef Kaucky´) Evaluation of a class of binomial coefficient
    summations, J. Combinatorial Theory, 1(1966), 233-247. Errata, ibid.,
    Ser. A, 12(1972), 309-310. (Cf. #82.2). MR 35(1968), #1493.
  52. Note on a q-series transform, J. Combinatorial Theory, 1(1966), 397-399.
    MR 34(1967), #1745.
  53. (with C. A. Church, Jr.) Lattice point solution of the generalized problem
    of Terquem and an extension of Fibonacci numbers, Fibonacci Quart.,
    5(1967), 59-68.  MR 37(1969, #133.
  54. A bibliography of research in mathematics at West Virginia University,
    1867-1967, Proc. W. Va. Acad. Sci., 38(1966), 123-133 (1967).
  55. Note on a combinatorial identity in the theory of bi-colored graphs,
    Fibonacci Quart., 5(1967), 247-250. MR 36(1968), #5024.
  56. History of the old telescope and observatory at West Virginia University,
    Proc. W. Va. Acad. Sci., 39(1967), 1-8 (1968).
  57. The bracket function, q-binomial coefficients, and some new Stirling
    number formulas, Fibonacci Quart., 5(1967), 401-423. Errata, ibid.
    6(1968), 59. MR 37(1969), #1262.
  58. Some remarks on backward pairs, Mathematics Notes, Washington State
    Univ., 11(1968), No. 4, Nov., 1-2.
  59. Note on a paper of Paul Byrd, and a solution of problem P-3, Fibonacci
    Quart., 6(1968), 318-321. MR 39(1970), #4465.
  60. Note on two binomial coefficient identities of Rosenbaum, J. Math.
    Physics, 10(1969), 49-50. ZBL 179(1970), p. 318. RZ 1969, #8B159.
  61. The bracket function and Fontené-Ward generalized binomial coefficients
    with application to Fibonomial coefficients, Fibonacci Quart., 7(1969),
    23-40, 55. MR 39(1970), #4021.
  62. Power sum identities for arbitrary symmetric arrays, SIAM J. Applied
    Math., 17(1969), 307-316. MR 40(1970), #52.
  63. Invariance and other simple techniques as a genesis of binomial
    identities, Nieuw Archief voor Wiskunde, (3)17(1969), 214-217.
    MR 41(1971), #78.
  64. (with C. N. Cochran) Finding squares in recurrent sequences with a desk-
    top computer, Monograph, Morgantown, W. Va. 1970. xvi + 50 pp.
    (iii + 13 pp. text + 50 pp. tables).
  65. Astronomy's might-have-been years, A history of the "oldest science" at
    WVU, West Virginia University Magazine, Vol. 3, No. 2, Summer, 1971,
    18-23.
  66. Remarks on a paper of Srivastava, J. Mathematical Physics, 12(1971),
    1378-1379. MR 44(1972), #2622.
  67. Comment on a paper of Chiang, J. Mathematical Physics, 12(1971), 1743.
  68. Noch einmal die Stirlingschen Zahlen. Bermerkungen zu Noten von
    Johannes Thomas und F. Klein-Barmen, Jber. Deutsch. Math.-Verein.,
    73 (1971), 149-152. MR 57(1979), #16093.
  69. Equal products of generalized binomial coefficients, Fibonacci Quart.,
    9(1971), 337-346. MR 45(1973), #6641.
  70. Explicit formulas for Bernoulli numbers, Amer. Math. Monthly, 79(1972),
    44-51. MR 46(1973), #5229.
  71. (with J. Kaucky´) Corrigendum on: "Evaluation of a class of binomial
    coefficient summations", J. Combinatorial Theory, Ser. A, 12(1972),
    309-310. (Cf. #52). MR 45(1973), #3217
  72. A new symmetrical combinatorial identity, J. Combinatorial Theory,
    Ser. A, 13(1972), 278-286. MR 46(1973), #3328.
  73. A new primality criterion of Mann and Shanks and its relation to a
    theorem of Hermite, Fibonacci Quart., 10(1972), 355-364; 372. Errata,
    ibid. 10(1972), 656. MR 47(1974), #119.
  74. The case of the strange binomial identities of Professor Moriarty,
    Fibonacci Quart., 10(1972), 381-391; 402. Errata, ibid., 10(1972), 656.
    MR 48(1974), #10838.
  75. A new greatest common divisor property of the binomial coefficients,
    Fibonacci Quart., 10(1972), 579-584, 628. Errata, ibid., 11(1973), 168.
    ZBL 256(1973), #10001, RZ 1973, #8B293.
  76. Some combinatorial identities of Bruckman - A systematic treatment
    with relation to the older literature, Fibonacci Quart., 10(1972), 613-
    627. Errata, ibid., 11(1973), 168. MR 48(1974), #75.
  77. A triangle with integral sides and area, Fibonacci Quart., 11(1973),
    27-39. Errata, ibid., 11(1973), 294. MR 46(1973), #8970.
  78. A remarkable combinatorial formula of Heselden, Proc. W. Va. Acad. Sci.,
    43(1971), 198-201 (1973). MR 50(1975), #4327.
  79. (with C. N. Cochran) Finding squares in recurrent sequences with a desk-
    top computer, Proc. W. Va. Acad. Sci., 43(1971), 202-208 (1973).
    MR 47(1974), #6601.
  80. Two remarkable extensions of the Leibniz differentiation formula,
    Bul. Inst. Politehn. Iasi, New Series, Sect. I, Math., Theoret. Mech., Physics,
    18(22)(1972), 47-53. MR 48(1974), #2314.
  81. Some combinatorial identities: Comment on a paper of Srivastava,
    Nordisk Matematisk Tidskrift, 21(1973), 7-9. MR 48(1974), #3746.
  82. Evaluation of the finite hypergeometric series F(-n,1/2,j+1;4),
    Bull. Soc. Math. Phys. Macédoine, 22(1971), 11-17 (1973).
    MR 48(1974), #641; ZBL 282(1975), #33003. Note later corrected paper.
  83. Coefficient identities for powers of Taylor and Dirichlet series, Amer.
    Math. Monthly, 81(1974), 3-14. MR 49(1975), #4795.
  84. An expansion of the operator (x^(p+q)Delta^n f(x), Glasnik Matematicki,
    (3)8(28)(1973), 259-272. MR 48(1974), #6736.
  85. (with L. C. Hsu) Some new inverse series relations, Duke Math. J.,
    40(1973), 885-891. MR 49(1975), #2421.
  86. Generalization of Hermite's divisibility theorems and the Mann-Shanks
    primality criterion for s-Fibonomial arrays, Fibonacci Quart., 12(1974),
    157-166. MR 50(1975), #1897.
  87. Note on functions studied by LeVan, Chawla, and Euler, J. Natur. Sci. & Math.,
    Lahore, Pakistan, 12(1972), 421-430 (1974). MR 50(1975), #218.
  88. (with Jay R. Byerly) Mathematical analysis of Hammer zones used in
    terrain corrections for gravimeter stations, Proc. W. Va., Acad. Sci.,
    43(1973), 284-289 (1974).
  89. A note on the number of linearly inducible orderings of points in d-space,
    SIAM J. Applied Math., 26(1974), 528-530. MR 49(1975), #4796.
  90. The design of the four binomial identities: Moriarty intervenes,
    Fibonacci Quart., 12(1974), 300-308. MR 50(1975), #9609.
  91. Improved evaluation of the finite hypergeometric series F(-n,1/2,j+1;4),
    Proc. W. Va. Acad. Sci., 45(1973), 317-323 (1974). MR 50(1975), #5045.
  92. Some sequences generated by spiral sieving methods, Fibonacci Quart.,
    12(1974), 393-397. ZBL 305(1976), #10006. RZ 1975, #7A163.
  93. Formal proof of equivalence of two solutions of the general Pascal
    recurrence, Fibonacci Quart., 13(1975), 127-128. MR 50(1975), #9772.
  94. Remarks on some formulas of Trumper, Fibonacci Quart., 14(1976),
    143, 146. Published as a Letter to the Editor. MR 54(1977), #2577.
  95. Types of orthogonality for arrays of numbers, Proc. W. Va. Acad. Sci.,
    47(1975), 219-228 (1977). ZBL 365(1978), #15013.
  96. A Fibonacci formula of Lucas and its subsequent manifestations and
    rediscoveries, Fibonacci Quart., 15(1977), 25-29. MR 55(1978), #126.
  97. A rearrangement of series based on a partition of the natural numbers,
    Fibonacci Quart., 15(1977), 67-72. MR 55(1978), #2740.
  98. Estimation of actuarial functions arising in coal mine valuation,
    Proc. W. Va. Acad. Sci., 49(1977), 67-83. ZBL 366(1978), #40001.
  99. Generalization of a formula of Touchard for Catalan numbers,
    J. Combinatorial Theory, Ser. A, 23(1977), 351-353. MR 56(1978), #8376.
  100. (with V. E. Hoggatt, Jr. and J. B. Kim) Sequences associated with t-ary
    coding of Fibonacci's rabbits, Fibonacci Quart., 15(1977), 311-318.
    ZBL 398(1979), #05027. RZ 1978, #10b658.
  101. Euler's formula for n-th differences of powers, Amer. Math. Monthly,
    85(1978), 450-467. MR 58(1979), #256. (to L. Carlitz on his 70-th birthday)
  102. Generalization of a formula of Hilden and Fenger, Nordisk Matematisk
    Tidskrift, 24(1976), 109-110 (1977). Summary on p. 135. MR 58(1979),
    #255.
  103. Estimation of actuarial functions arising in coal mine valuation,
    Virginia Journal of Science, 29(1978), Mo.1, Spring, 10-17. An amended
    version of No. 97 (stressing examples of valuations in a new Sect. 9). See
    also Abstract in Va. J. Sci., 28(1977), No.2, 58. MR 80h: #90008.
  104. Evaluation of sums of convolved powers using Stirling and Eulerian
    numbers, Fibonacci Quart., 16(1978), 488-497, 560-561. MR 80b: #05007.
  105. Operational formulas for unusual Fibonacci series, Fibonacci Quart.,
    16(1978), 555-560. MR 80b: #05008.
  106. A history of the Fibonacci Q-matrix and a higher-dimensional problem,
    Fibonacci Quart., 19(1981), 250-257. MR 82j: #01055.
  107. (with M. E. Mays) Series expansions of means, J. Math. Anal. and Appl.,
    101(1984), 611-621. MR 86a: #26025.
  108. (with W. E. Greig) A Lucas triangle primality criterion dual to that of
    Mann-Shanks, Fibonacci Quart., 23(1985), 66-69. MR 86h: #11007.
  109. New inverse series relations for finite and infinite series with
    applications, J. Math. Research and Exposition, 4(1984), 119-130.
    MR 87k: #40003.
  110. (with Aldo Peretti) The transformation of series into products and
    viceversa, Bull. Number Theory and Related Topics, Buenos Aires,
    9(1985), Dec., No.3, 13-15. RZ 1987, #9b16.
  111. The g-inverse and reverse-converse series inverse, Indian J. Pure and
    Appl. Math., 17(1986), 613-628. MR 87m: #05019.
  112. Number-theoretic identities involving the Möbius function and the
    logarithm, Indian J. Math., Ramanujan Centenary Vol., 29(1987),107-115.
    MR 89d; #11003.
  113. (with M. E. Mays) Counting chains of numbers, Utilitas Mathematica,
    31(1987), 227-232. MR 88h: #05006.
  114. Some formulas for Euler's constant, Bull. Number Theory and Related
    Topics, Buenos Aires, 10(1986), No.2, 2-9 (1988). MR 89g: #11011.
  115. (with W. Y. Kerr) Characterization of polynomial systems satisfying
    generalized convolution differential-difference equations,
    J. Math. Research and Exposition, Dalian, PRC, 9(1989), No. 2, May,
    159-172. MR 90f: #34104.
  116. Equivalence of Piza's primality criterion with that of Gould-Greig and
    its dual relationship to the Mann-Shanks criterion, Fibonacci Quart.,
    27(1989), No.4, August, 362-365. MR 91g: #11005.
  117. (with H. Gingold and M. E. Mays) Power product expansions, Utilitas
    Mathematica, 34(1988), 143-161. MR 90i: #40003.
  118. Series transformations for finding recurrences for sequences, Fibonacci
    Quarterly, 28(1990), No. 2, May, 166-171. MR 91b: #11016.
  119. A bracket function series transform and its inverse, Fibonacci Quarterly,
    32(1994), No. 2, May, 176-179. MR 95e: #11031.
  120. A class of binomial sums and a series transform, Utilitas Mathematica,
    45(1994), May, 71-83. MR 95c: #05006.
  121. (with Paula Schlesinger) Extensions of the Hermite g. c. d. theorems
    for binomial coefficients, Fibonacci Quarterly, 33(1995), 386-391.
    MR 97g: #11015
  122. A set of polynomials associated with the higher derivatives of y = x^x,
    Rocky Mountain J. Math., Vol. 26(1996), 615-625. MR 97e: #26012.
  123. (with H. M. Srivastava) Some combinatorial identities associated with the
    Vandermonde convolution, Applied Mathematics and Computation
    (Elsevier Science Publ.), 84(1997), No. 2-3, 97-102. MR 98d: #05016.
  124. (with Temba Shonhiwa) Functions of GCD's and LCM's, Indian Journal
    of Math. (Allahabad), 39(1997), No.1, 11-36. MR 98j: #11001
  125. (with Temba Shonhiwa) A generalization of Cesáro's function and other
    results, Indian Journal of Mathematics (Allahabad), 39(1997), No. 2,
    183-194 (1998). MR 99f: #11010
  126. In Memoriam, Paul Erdös, 1913-1996,  Journal of Mathematical Research
    and Exposition, Dalian, China, 17(1997), No.1, Feb., 1-4. With Chinese
    translation tby Mr. Xiaodong Shi,WVU math. graduate student.
  127. The Girard-Waring power sum formulas for symmetric functions, and
    Fibonacci numbers, Fibonacci Quarterly, 37(1999), No. 2, May,
    pp. 135-140. (Dedicated to L. Carlitz on his 90-th birthday) MR 2000b: #11014
  128. (with J. Wetweerapong) Evaluation of some classes of binomial identities
    and two new sets of polynomials, Indian Journal of Mathematics, 41(1999),
    No.2, pp. 159-190. (invited paper for the B. N. Prasad Birth Centenary
    Commemoration Volume). MR 2001b: #05018.
  129. The generalized chain rule of differentiation with historical notes,
    Utilitas Mathematica, 61(2002), May, 97-106. MR 2003a:#26007
  130. Annihilation coefficients, in "Analysis, Combinatorics and Computing,''
    (Tian-Xiao He, Peter J. S. Shiue, and Zhongkai Li (eds.)),
    Nova Science Publishers, Inc., 2002, 205-223. MR 2005g: #05021.
    (80-th Birthday Conference in Honor of Prof. L. C. Hsu, Dalian, China,
    August 2000)
  131. The Mann-Shanks primality criterion for m-nomial arrays,
    Bulletin of the Institute for Combinatorics and its Applications,
    37(2003), Jan. 15-18. MR 2003k: #11189.
  132. Proof and generalization of a Catalan number formula of Larcombe,
    Congressus Numerantium, 165(2003), 33-38. MR 2004m: #05023.
  133. A curious identity which is not so curious. Mathematical Gazette
    (London, England 88(2004), Issue No. 511, March, p. 87.
  134. Sums and convolved sums of Catalan numbers and their generating
    functions, Indian Journal of Math. 46(2004), Nos. 2-3, 137-160.
  135. Sums of powers of binomial coefficients via Legendre polynomials,
    Ars Combinatoria, 73(2004), 33-43. MR 2006a: #11019.
  136. (with Steve Fairgrieve) Product Difference Fibonacci Identities of
    Simson, Gelin-Cesàro, Tagiuri and generalizations, Fibonacci Quarterly, 
    43(2005), pp.137-141. MR 2005m:#11019. (to Sam B. Nadler, Jr. on his 65-th birthday)
  137. Remarks on Compositions of Numbers into Relatively Prime Parts,
    Notes on Number Theory and Discrete Mathematics, 11(2005), No. 3. pp.1-6.
  138. Classes of iterated binomial transforms, Indian J. Math., 47(2005), No.2-3,
    pp.183-203. MR 2007i: #39033.
  139. A baker's dozen proofs of a binomial identity, Publikacije Elektrotehnickog
    Fakulteta, Ser. Mat., 16(2005), 131-138. MR 2006c: #05025.
  140. Curious special function identities. (A generalization of No. 131 above)
    Publikacije Elektrotehnickog Fakulteta (Univ. Beograd), Ser. Mat., 18(2006),
    22-27. MR 2008b: #05016
  141. The inverse of a finite series and a third-order recurrent sequence,
    Fibonacci Quarterly, 44(2006), Nov. 302-315.  MR 2008f: #11016.
  142. (with the late L. Carlitz), Remarks on MacMahon's identity for sums of cubes of
    binomial coefficients, Ars Combinatoria, Vol. 85(2007), 221-224. MR 2008i: #05005.
  143. Higher order extensions of Melzak's formula, Utilitas Mathematica,
    Vol. 72(2007), 23-32. MR 2008b: #05009.
  144. Note on a combinatorial formula of Rosenberg, Bulletin of the Institute
    of Combinatorics and Its Applications, Vol. 51(2007), Sept. pp. 5-12.
  145. (with Jocelyn Quaintance) A linear binomial recurrence and the Bell numbers
    and polynomials, Applicable Analysis and Discrete Mathematics (Belgrade),
    Vol.1(2007), No. 2, 371-385.  MR 2008g: #05014
  146. (with Jocelyn Quaintance) Additional analysis of binomial recurrence
    coefficients, Applicable Analysis and Discrete Mathematics,Vol.1(2007), 
    No. 2, 386-396.  web: http: //pefmath.etf.bg.ac.yu     MR 2008h: #05012
  147. (with Jocelyn Quaintance) Floor and roof function analogs of the Bell numbers,
    Integers Journal, Vol. 7(1)(2007),  Paper # A58, 21 pages. MR 2008m: #11053.
    Erratum published 10 Dec. 2009 as a PDF available with the paper on INTEGERS on
    Vol. 7 website.
  148. Sums of powers of binomial coefficients via Legendre polynomials, Part 2,
    Ars Combinatoria, Vol. 86(2008), Jan. 161-173. MR 2008m: #33022.
  149. A note on Combinatorial Identities arising from the Lagrange-Waring
    Interpolation Formula, Ars Combinatoria, Vol, 86(2008), 281-288. MR 2008m: #05029.
  150. (with Temba Shonhiwa) A Catalogue of Dirichlet Series,  Missouri Journal
    of Mathematical Sciences, Vol. 20(2008), No. 1, Winter issue, second paper, 
    17 pages. Online at: http://www.math-cs.ucmo.edu/~mjms/current.html. 
  151. (with Jocelyn Quaintance) Products of numbers which obey a Fibonacci-type
    recurrence, Fibonacci Quarterly, 45(2007), Nov., No. 4, 337-346. Publ. Dec. 2008.
    MR2009j: #11029.
  152. (with Jocelyn Quaintance) Implications of Spivey's Bell number formula, Journal
    of Integer Sequences, 11(2008), Issue 3, paper no. 7, 6pp. MR 2009h: #11033.
  153. Proof and generalization of an Identity of E. T. Bell, Bulletin of the Allahabad Math.
    Society, 24(2009), Part 1,  pp.  71-82,  
  154. (with Jocelyn Quaintance) Partial fraction expansions and a question of Bruckman,
    Fibonacci Quarterly, Vol. 46/47, No. 3, Aug. 2008/2009, pp. 245-248. MR 2010i: #05035.
  155. (with Jocelyn Quaintance) Bell Numbers and Variant Sequences Derived from a
    General Functional Differential Equation, Integers Journal, 9(2009), paper #A44, pp. 581-589.
  156. (with Jocelyn Quaintance) Operators, inverse series relations and other expansions connected
    with generalized Gould-Hopper polynomials, Indian J. Math., (2009), 51(2009), No. 3, Dec.
    pp. 463-487.
  157. Kai Lai Chung (1917-2009): In Memoriam, Journal of Mathematical Research and Exposition,
    (Dalian, China) Vol. 29(2009), No. 6, pp. 952-953.
  158. (with Jocelyn Quaintance) Generalizations of Vosmansky's identity, Fibonacci Quarterly, 
    Vol. 48(2010), No. 1, Feb. pp. 56-61.
  159. (with Jocelyn Quaintance) Annihilation coefficients, binomial expansions and q-analogs,  Journal
    of Mathematical Research and Exposition, Vol. 30(2010). No. 2, pp. 191-204.
  160. (with Salar Y. Alsardary) Special function inverse series pairs, Kyungpook Mathematical Journal,
     Vol. 50(2010), No. 2, pp. 155-193.
  161. (with Jocelyn Quaintance) Inverting a finite series with constant coefficients, Fibonacci Quarterly,  49(2011), May, No. 158-165
  162. (with Timothy J. Glatzer) Catalan and Bell Numbers, Research Bibliography,
    New on-line edition 2011. Download PDF here: BibGood final.pdf
  163. (with Jocelyn Quaintance) A Ballot Number Formulary, Ars Combinatoria, 
    Vol. 102(2011),  365-383.  
  164. (with Jocelyn Quaintance) A one parameter generalization of Bell's sum, Indian Journal of
    Mathematics, Vol.53(2011), No. 2, August, pp. 301 -329.
  165. (with Jocelyn Quaintance) Double fun with double factorials, Mathematics Magazine, 85(2012), No. 3, June, pp. 177-192. 
  166. (with Joel V. Brawley and John Brillhart) Recollections of Leonard Carlitz, Acta Arithmetica, 152(2012), 361-372. 
  167. (Joel V. Brawley, John Brillhart, and Henry W. Gould, Editors) Publications of Leonard Carlitz, Acta Arithmetica, 152(2012), 373-405 (annotated catalog of Carlitz's 773 publications)
  168. (with Tian-Xiao He) Characterization of (c)-Riordan Arrays, Gegenbauer-Humbert-type polynomial sequences, and (c)-Bell polynomials. J. Math. Research with Applications, 33(2013), No. 5, 505-527. 
  169. (with Jocelyn Quaintance) Bernoulli numbers and a new binomial transform identity, Journal of Integer Sequences, 17(2014), Issue  No..2, Article 14.2.2.  
  170. Catalan Numbers and the Protean nature of binomial coefficient notation,  Bulletin of the Institute for Combinatorics and its Applications, Vol. 71(2014), May, pp. 77-86.     Catalan Memorial Issue.  

Some Miscellaneous Book Reviews, Letters, Newspaper Articles, Abstracts, Problem proposals and solutions. etc.

  1. What is mathematics?  A newspaper series in three parts: Chap. 1, Dominion-News Magazine, Morgantown, W.Va., Vol.4, No. 35, Sat. 2 Sept. 1961, p.2; Chap. 2, loc. cit., Vol.4, No. 36, Sat. 9 Sept. 1961, p.2; Chap. 3, loc. cit., Vol.4, No. 37, Sat.16 Sept. 1961, p.2.
  2. The Fibonacci Numbers as a key to enthusiasm, Math. and Science Newsletter for Secondary Schools, Charleston, W.Va., Vol.1-S(1963), Dec.,. No.1, 10-11.
  3. Book Review of: 1962 Chelsea reprint of 'Introduction to the Operations with Series' by I. J. Schwatt; Amer. Math. Monthly, 70(1963), 107.
  4. Book Review of: 'Operational Calculus and Generalized Functions' by A. Erdélyi, N.Y., 1962; Amer. Math. Monthly, 71(1964), 574.
  5. Review of:  H. W.Gould, "Conguences involving sums of binomial coefficients and a formula of Jensen", Amer. Math. Monthly, 69(1962), 400-402, (Cf. #28), Math. Rev., 38(1969), #2033.
  6. Review of: David Blackwell & Lester Dubins, "An elementary proof of an identity of Gould's", Bol. Soc. Mat. Mexicana, (2)11(1966), 108-110, Math. Rev., 38(1969), #2034.
  7. Letter to the Editor, Notices of Amer. Math. Soc., 16(1969), No. 4, June, p. 624.  Concerns upside-down Aleph font in Russian journals.
  8. Book Review of: 'A History of π (Pi)', by Petr Beckman, Sec. Ed., Boulder, Colorado, 1971; Math. of Computation, 28(1974), 325-327. 
  9. Letter to DEAR ABBY, Published in Dominion-Post and numerous other newspapers nationwide, Mon. 6 May 1974.    Concerns a certain solution to the three ancient problems of squaring the circle, trisecting the angle, and duplicating the cube. Discusses work of Edward V. Graef and V. C. Harris.
  10. Derivation of q-identities from the Lagrange interpolation formula, from q-Stirling numbers, and from other sources, Abstract in: Eulerian Series and Applications, Abstracts and Problems from the Conference held at Penn. State Univ., 15-17 May 1974.
  11. Book Review of: 'Die mathematischen Studien von G. W. Leibniz zur Kombinatorik' by Eberhard Knobloch, Wiesbaden, 1973; ISIS 67(1976), 315-316. 
  12. Book Review of: 'Die mathematischen Studien von G. W . Leibniz zur Kombinatorik. Textband' by Eberhard Knobloch, Wiesbaden, 1976; ISIS 71(1980), 178.
  13. Book Review of: 'Der Beginn der Determinantentheorie: Leibnizens nachgelassene Studien zum Determinantenkalkul. Textband', by Eberhard Knobloch, Hildesheim, 1980; ISIS, 73(1982), 119. 
  14. Which numbers are interesting?, The Mathematics Teacher, 73(1980), 408. Letter to the Editor.
  15. LAFF GRAFFS, A Contribution to Mathematical Humor and Wit, published
    by the author, Morgantown, 1987.  i + 15pp.
  16. A critical view: The new editor looks at journal standards, Proc. W. Va. Acad. Sci., 46(1974),
    No.1, xviii-xxi (1975).    Editorial.
  17. Outgoing editor's closing remarks, Proc. W. Va. Acad. Sci., 50(1978), pp. x-xiii (1980).
  18. In Memoriam, Paul Erdös, 1913-1996,  Journal of Mathematical Research and Exposition, Dalian, China, 17(1997), No.1, Feb., 1-4. With Chinese translation by Mr. Xiaodong Shi, WVU math. graduate student.
  19. Solution of Prob. 4211, Amer. Math. Monthly, 63(1956), 126-127, Ref. to paper of mine, pp. 84-91.
  20. Solution of Prob. 375, Math. Mag. 33(1959), 115-117. A sum posed by D. A. Steinberg.
  21. Problem E 1388, Amer. Math. Monthly, 66(1959), 813. Soln. by N.J. Fine, ibid. 67(1960), 476.
  22. Solution of Prob. 470, Math. Mag. 36(1963), 206. Posed by W. Squire.
  23. Solution of Prob. 505, Math. Mag., 35(1963), 267-269. Posed by Carlitz.
  24. Solution of Prob. 521, Math. Mag., 37(1964), 61. Posed by  L. Moser.
  25. (with T. A. Chapman) Solution of functional equations involving Túran expessions,
    Proc. W. Va. Acad. Sci., 35(1963), 205(1964). Abstract.
  26. Properties of polynomials associated  with expansions of certain difference operators,
    Proc. W. Va. Acad. Sci., 35(1963), 208(1964). Abstract.
  27. Solution of Problem 548, Math. Mag., 37(1964), 357-358. Posed by L. Moser.
  28. Problem 5231, Amer. Math. Monthly, 71(1964), 923. Solns by Klamkin, Carlitz,  and Riordan,
    ibid., 72(1965), 921-923.
  29. Associativity and the golden mean, Fibonacci Quarterly, 2(1964), 203.
  30. Prob. E 1767, Amer. Math. Monthly, 72(1965), 315. Soln by E. S. Langford,
    ibid., 73(1966), 540. 
  31. Angle, article in the New Standard Encyclopedia, Standard Education Society, Chicago,1965,
    Vol. A, pp. 257-258. Note credits, p. xix.   (written in 1955-56.
  32. Solution of Prob. 571, Math. Mag., 38(1965), 185-187.
  33. A Fibonacci crossword puzzle, Fibonacci Quart., 4(1966), 59-61; Soln, ibid., p.150. 
  34. Comment on Prob. 602, Math. Mag., 40(1967), 107-108. Posed by Bruce W. King.
  35. (with Maurice Machover) Problem E 1975, Amer. Math. Monthly, 74(1967), 437-438.
    Solution by M. T. L. Bizley, ibid. 75(1968), 682-683.
  36. Solution of Prob. 67-5, SIAM Review, 10(1968), 225-226. Posed by D. J. Newman
    W. Weissblum.
  37. Late comment on Prob. 67-14, SIAM Review, 10(1968), 453. (inverse of Newton-Girard
    formula)
  38. (with Franklin C. Smith) Problem 5612, Amer. Math. Monthly, 75(1968), 791.
    Soln by M. T. L. Bizley, ibid., 76(1969), 705-707.
  39. Solution of Prob. 5541, Amer. Math. Monthly, 72(1968), 1019-1021. Posed by Robert Breusch.  Ramanujan's exponential limit.
  40. Comment on Prob. 612, Math. Mag., 39(1966), No. 4, 253. Joint comment with William Squire. 
  41. Solution of Prob. 676, Math. Mag., 41(1968), 165-166. Posed by William Squire.
  42. Solution of Prob. 5463, Amer. Math. Monthly, 75(1968), 204-205. Posed W. G. Spohn.
  43. Problem H-142, Fibonacci Quart., 6(1968), 252. Soln by L. Carlitz, ibid. 8(1970), 275-276.
  44. Power sum identities for arbitrary symmetric arrays, Proc. W. Va. Acad. Sci., 40(1968),
    364 (1969).  Abstract.
  45. Problem 5777*, Amer. Math. Monthly, 78(1971), 202.   Extension of Hermite's g.c.d. theorems
    for binomial coefficients. 
  46. Solution of Prob. 783, Math. Mag., 44(1971), 289-291, Posed by H. L. Krall.
  47. Solution of Prob. 70-21, SIAM Review, 14(1972), 166-169. Posed by V. K. Rohatgi.
  48. Problem 825, Math. Mag., 45(1972), 100-101. Solns by V. Konecny and C. W. Trigg,
    ibid., 46(1973), 45-46.
  49. Comment on a paper of Wiggins on an occupancy problem, Math. Mag., 45(1972), in a paragraph 
  50. Problem E 2384*, Amer. Math. Monthly, 79(1972), 1034. Posed as a complement to E 2383. Comments, ibid., 81(1974), 170-171; Soln by Stuart Clary, ibid., 83(1976), 285-288.
  51. Problem E 2395, Amer. Math. Monthly, 80(1973), 75.  Posed as a complement to Prob. E 2383. Soln by St. Olaf Problem Group, ibid., 80(1973), 1146.
  52. Solution of Prob. 72-22, SIAM Review, 15(1973), 798-799. Posed by Barry Wolk.
  53. (with David Shelupsky) Problem E 2472, Amer. Math. Monthly, 81(1974), 406. Solns by Paul S. Bruckman and Otto G. Ruehr, ibid., 82(1975),525-526.
  54. Soln. of Prob. H-219, Fibonacci Quart., 13(1975), 185-187. Posed by Paul Bruckman.  (Solution generalizes a binomial identity) 
  55. Solution of Prob. 974, Math. Mag., 50(1977), 216. Posed by John P. Hoyt.
  56. Remark on Prob. H-123, Fibonacci Quart., 16(1978), 189. Posed by Douglas Lind.
  57. Solution and generalization of Problem 11343 [2008, 166], American Math. Monthly 116(2009), Dec., pp. 944-945. Generalization by Gould and Quaintance noted where we expand the n-th power of a quadratic in two ways.
  58. About Fibonacci Numbers, posted as a Facebook Note, Feb. 2011.
  59. Bell and Catalan Numbers, posted as a Facebook Note, 20 March 2011.


Memorial items

     1. Wilbur Harris Bluhm (1920-2008), In Memoriam. Written 20 May 2009. Read in part
          at memorial service at St. Mary's Catholic Church, 2 June 2009.
     2. Donald M. Bondurant (1921-2003) - In Memoriam,
     3. Kai Lai Chung (1917-2009): In Memoriam, JMRE, Vol. 29(2009), No. 6, pp. 952-953.
         Also posted on my web page.
     4. Allen B. Cunningham, In Memoriam, written 7 May 1991.
     5. In Memoriam, Paul Erdös, 1913-1996,  Journal of Mathematical Research and Exposition, Dalian,
         China, 17(1997), No.1, Feb., 1-4. With Chinese translation by Mr. Xiaodong Shi, WVU math.
         graduate student.
     6. Roger Mersing (1941-2009), In Memoriam. Read at funeral, 15 August 2009.
     7. Temba Shonhiwa (1958-2009): In Memoriam. Posted on my web page, 10 Nov. 2009.
     8. William (Bill) Squire, Remarks at retirement luncheon, 9 April 1986.
     9. William (Bill) Squire (1920-2007), Mary Lurline Scott Squire (1917-2007),
          In Memoriam,  3 Nov. 2007. Extension of remarks made at a retirement
          luncheon for Bill, 9 April 1986.
     
     

Pending papers and papers in process

  1. A formula for sums of powers of numbers, written in 1945, to be revised.
  2. Sums and convolved sums of Rothe coefficients,
  3. Worpitzky's formula for Bernoulli numbers and generalizations,
  4. Differences of the harmonic series, Stirling numbers, and
    q-analogs, Being revised.
  5. (with J. Wetweerapong) q-analogs of some binomial identities.
  6. Generalized binomial theorem and identities,
  7. Persymmetric matrices and determinants of Catalan numbers,
  8. Persymmetric matrices and determinants of Bell numbers,
  9. A new inverse series relation motivated by the Bell numbers,
  10. A Short Catalog of q-Binomial Coefficient Identities.
  11. (with Steven L. Richardson and Andrea D. Sutyak) Catalog of Polygon
    Dissections, Circular Trees and Their Relation to Other Combinatorial and
    Number-theoretic Identities, April 2004.
  12. (with A. M. Chak) Expansions of Multidimensional Catalan Numbers,
  13. John Eiesland, the Mathematics Library and the Eiesland Desk,
    Remarks made at Dedication of the Eiesland Desk and Refurbished
    WVU Mathematics Library, Monday 26 April 1999,
  14. Historical note on the four-color conjecture, the work of C. N.
    Reynolds, and remarks about libraries and mathematics departments,
    (written 1992, revised 1997)
  15. Determinants of multifactorials.
  16. The coefficients in the series expansion of y = (1+1/x)^x.
  17. Novel combinatorial identities analogous to those of Vandermonde, Part 1,
  18. Novel combinatorial identities analogous to those of Vandermonde, Part 2,
  19. Howard Eves and other great mathematical catalysts who made all things possible,
  20. Some binomial identities of Munarini, 
  21. Three combinatorial identities of Seiffert,
  22. An integration trick, (pedagogical note)
  23. An identity of Oliver Lung-Feng Chen,
  24. Some properties of the Bessel polynomials of Emil Grosswald,
    or: Generalizations of binomial identities of Carlitz, Grosswald and
    Riordan. (Cited by Grosswald, Bessel Polynomials, Lecture Notes in Math.,
    Vol. 698, Springer, N.Y., 1978, p. 57.)
  25. Sierpinski's criterion for rational numbers,
    (Begun in 1960's with graduate assistant John Skocik at WVU)
     with extensive tables computed by Addison M. Fischer,
  26. An extension of the Truesdell F-equation, long memoir,
  27. Fibonomial Catalan numbers: Arithmetic properties and a table of the
    first fifty numbers, 
  28. A Catalogue of 100 Postulate Sets for Boolean Algebra, (Begun c. 1954
    in collaboration with the late Henry Warren Thwing at U.Va.),
  29. Proofs of some theorems in Boolean algebras. written circa 1954,
  30. The moments of a distribution as a genesis of combinatorial
     identities, (inspired by remarks of Vincent A. Uthoff at WVU;
     See formulas (4.22)-(4.29) in my book Combinatorial Identities),
  31. (with J. B. Kim) On matrix semigroups and a q-function related to
     theta functions,
  32. (with Hollie L. Buchanan) Legal numbers, a problem in permutations,
  33. Some  L. C. M.  properties of the binomial coefficients, in collaboration
    with Paul Erdös, and assisted by Arsen Melkumian,
  34. Some new inverse series relations and new formulas for Stirling
    numbers, Cf. Notices of Amer. Math. Soc., 21(1974), p. A-603,
  35. The derivation of operational expansions by orthogonality relations,
     Cf. article "The Case for Vectors" by Elsa Nadler, Inquiry, WVU,
     Spring 1986, pp. 25-27, where a very brief account of some of this
     was recounted.
  36. Note on a paper of Stalley concerning a generalized geometric series,
    (written in 1953; submitted to Monthly; returned for revision; Carlitz
     was evidently the referee),
  37. (with Michael Caulfield and Cheng Zhao) Operators, inverse relations
     and other expansions connected with generalized Gould-Hopper
     polynomials, This has become a joint paper with Dr. Quaintance.
  38. Stirling number, binomial coefficient, and summation notations,
    and some personal reminiscences of mathematics, 1992; 2003
  39. Professional self-management: A personal case history. Invited paper
    to be submitted to a refereed journal on behaviorology. written for a
    WVU Educational Behaviorology Seminar in 1991; rev. 2000; 2009.
  40. Remarks concerning Euler's constant and a collection of 100 formulas
    for this number, 
  41. Generalized Bernoulli and Euler polynomial convolution identities.
    written Dec. 1994, revised 2001. 
  42. Riordan's proof of Grosswald's identity,
  43. Some Combinatorial identities of Brodzki,
  44. Some trees in my forest,
  45. Mathematical Identities, An Investigation into the Nature of Research,
  46. History of Combinatorial Identities. 
  47. The Magic Eleven and the Lucas Team Identities, joint endeavor with Mike Long,
  48. Alice in Boola-Boola-Land, with apologies to Lewis Carroll,
  49. Bibliography on the chain rule of differentation,
  50. Combinatorial identities derived from powers of 2 x 2 matrices,
  51. The origin, evolution and naming of two sciences and their practitioners,
  52. The emergence of combinatorics as a separate mathematical discipline,
     and the parallel case of behaviorology versus psychology.
  53. Notes on Fibonacci numbers, written in May 1979, expository.
  54. (with David C. Rine & W. L. Scharff), Algorithm and computer program
    for the determination of equal products of generalized binomial
    coefficients,  started in 1972 . . . cited in No. 75 (new g.c.d. property).
  55. The origin and application of combinatorial identities, Benedum
    Distinguished Scholar Award lecture, 29 March 1988. Revised 31 Jan.
    2004.
  56. A series identity and Legendre polynomials of positive and negative
    index,
  57. The interchange of order of summation in double sums,
  58. Operations with series and formal power series: A tutorial paper, 1993
  59. The interchange of order of summation in double sums, Nov. 1962.
  60. Notes on reciprocal derivatives,  begun in 1948.
  61. A quadratic recurrence, paper begun with George E. Trapp, c. 1970's.
  62. A formula of Hayes for sums of logarithms of binomial coefficients,
  63. Lagrange interpolation and generalization of a theorem of Howard,
  64. (with the late Wilbur. H. Bluhm) "Fractionons", Fraction-sensitive means, 
    original version sent to College Math. Journal;  being revised.
  65. Factoring quadratics, to submit to a math. education magazine.
  66. (with the late Vadim Komkov) The log of a sum and symmetrical function
    expansions,
  67. (with T. A. Chapman) A finite Laplace transform,
  68. All angles are equal - analysis of a paradox given by Wentworth,
     (Classroom notes for analytic geometry; a formal paper was sent to
     The Mathematics Teacher, returned for revision) 
  69. How to wrap a package: Some notes on geodesics,
    (Have a cardboard model of these geodesics)
  70. A Euclidean geometric dissection problem,
  71. Remarks on the equation f'(ax) + f(x) = 0,
  72. Length of a spiral of uniform pitch,
  73. A congruence of Gegenbauer for number-theoretic sums,
  74. A bibliography on convolution relations and related series,
    A five-page list giving 58 references. Compiled Dec. 1957 at U.N.C.
  75. Waves of the future: Twenty-three year cycle of building and slowdown
     in hiring of mathematics research faculty at WVU: evidence, causes,
     implications for planning.    An application of the logistics curve. 
  76. Bio-bibliographical Chronicle of Faculty and Student Publications in
    Mathematics at West Virginia University, 1867 - present,
    Data for this is stored on 3000 index cards and in other papers.
  77. How to choose a calculus book . . . by weight!, 25 pp. statistical tables, 1986.
    In collaboration with the late Prof. Wilbur H. Bluhm.
  78. Chromatic polynomials, 13 April 1987; revised 8 Dec. 1992; revised
    28 Oct. 1992.
  79. Some terminology suggested by matrix theory (linear algebra),
    24 Jan. 1991; Revised 15 Jan. 1993.
  80. Proof as convergence,
  81. A ternary Farey process, 
  82. Generalized trigonometry,
  83. Review of the Campbell-Baker-Hausdorff theorem, 13 page paper given
    at a colloquium in the WVU Math. Dept. Paper is dated 14 May 1971.
  84. (with Andrew R. Cogar, Melissa R. Ferrell and Jun Hua) Sum and
    difference unit fraction expansions, Written Feb.-March 1997. Sent to
    College Math. Journal; To be revised.
  85. Baldwin's 'trisection', a classroom note, written 27 - 29 March 1997.
  86. Two geometric constructions, trisection of an angle and a more
    general figure,   Quillen's trisection method,
  87. (with John H. Biggs) Comments on Presentation in Sigley and Stratton's
    'Solid Geometry', written 10 October 1960. (Math. 7 supplement)
  88. Mathematical Identities, An Investigation into the Nature of Research,
    Monograph notes for Math. 407, Fall 1997, WVU.   36 pp.
  89. A mathematical formula for the binding energy of all the isotopes
    of atomic elements 1 through 118. Formula developed c. 1945.
    unpublished. 
  90. 572957795130823208767981548141 . . . AND ALL THAT!  
    written 24 June 2005. Pedagogical note. Answers query from Bruce Welker.
  91. The atomic origins of my scientific curiosity, written 10 Jan. 2006
     and posted on my WVU Mathematics Department web site under
     'U235 Discovery.html'.
  92. Karl Friedrich Gauss and Benjamin Apthorp Gould, 9 July 2007. revised.
  93. Word Unprocessing, a new phenomenon, written 16 March 2009.
     posted on my WVU Math. web site.
  94. (with Timothy J. Glatzer) Further classes of iterated binomial transforms
    and related bracket function transforms, submitted.
  95. Limits of sums involving logarithms of binomial coefficients, 6 June 2006.
  96. Complex number extensions of combinatorial identities as a source of
    new combinatorial identities, given 13 Oct. 1993, Seminar, WVU
  97. A Bernoulli number identity of Carlitz and a new binomial transform, 
  98. Generalization of symmetrical binomial identities of Redl and Catalan,
  99. Generalized hypergeometric form of the Rothe expansions.
  100. (with Jocelyn Quaintance) A brief note on sums whose general term is
     a convolution, Under revision. 
  101. Note on a paper of Porter and identities of Abel type, written
    27 February 2007. Being revised. 
  102. (with Jocelyn Quaintance) The generalized Bell number recurrence
    and Catalan recurrence, 12 July 2007.
  103. (with Jocelyn Quaintance) Research Proposal: Catalan-Like Recurrences
    With Arbitrary Initial Conditions, 21 Jan. 2007.
  104. (with Jocelyn Quaintance) Sums of powers of binomial coefficients via Legendre
    polynomials, Part 3. February 2008, being revised.
  105. (with Jocelyn Quaintance) Applications of Legendre polynomials to derive
    combinatorial identities, In preparation, March 2008. 
  106. (with Jocelyn Quaintance) A combinatorial identity of Walther Janous, March 2008. 
  107. (with Jocelyn Quaintance) Interpolating the double factorial function, March 2009.
  108. (with Jocelyn Quaintance) A brief note on the differences of central binomial coefficients,
    March 2010.
  109. (with Jocelyn Quaintance) New recurrences for Stirling numbers, July 2011.
  110. (with Jocelyn Quaintance) A Bernoulli Number Identity Posed by Carlitz and a
    New Binomial Transform of Chen, written 15 Dec. 2011. Submitted, 2012
  111. (with Jocelyn Quaintance) Binomial identities of Ragnar Frish and Murray Klamkin,
     January 2012. Submitted.



H. W. Gould, Books - Monographs

  1. Combinatorial Identities, Second edition. Published by the author,
    Morgantown, WV, 1972. viii+106pp.  Copies available on request to the author.
  2. Combinatorial Identities, First edition. U. S. Department of Defense, Nov. 1959.
  3. Trigland, publ. by McGraw-Hill, 1995, ISBN 0-07-024498-7
    v + 37pp. Issued as a supplement to other McGraw-Hill trig books, such as
    Plane Trigonometry, fifth edition, by Gordon Fuller.
  4. Topics in Combinatorics, published by the author, Second edition, Lecture Notes.
    April 2000. iii+120pp. 
  5. Catalan and Bell Numbers: Research Bibliography of Two Special
    Number Sequences, fifth edition, 1979. (= Mathematica Monongaliae No. 12.)
  6. (with Timothy J. Glatzer) Catalan and Bell Numbers, Research
    Bibliography, New on-line edition 2011. Download PDF here: BibGood final.pdf
  7. Handbook of Arrays of Numbers, 1989. 150 pp. A collaboration begun
    c. 1976 with W. E. Greig. MacFortran software developed by
    Narad Maharaj. Only tentative manuscript available.  
  8. Ben Zion Linfield's Notes on the Geometry of the Triangle. Extension of
    notes for course given at University of Virginia in 1950's using vectors and
    matrices and barycentric calculus to study the geometry of the triangle. 100 pp. in process. 
  9. The Bateman Notes on Binomial Coefficients, book written in 1961, to be published.
  10. Handbook of Postulate Sets for Boolean Algebra, In preparation.
  11. H. W. Gould, Editor, with Donald M. Bondurant, Stephen A. Ford, and
     Michael M. Watts, Valuation Handbook, for Equitable ad valorem
     Taxation of Coal Property,  Monograph, xxviii + 720 pp. 1978. Unpublished.
     Concerns generalized Hoskold formula. Research supported by $16,000 WVU  Provost's grant.
  12. Mathematical Bibliography: A Guide to Information Retrieval in the Mathematical Sciences with  special emphasis on the libraries at West Virginia University, Aug. 1990, iii + 15 pp. 
    Revised Edition, West Virginia University, Aug. 1990, iii + 15 pp.
    Revised Edition, Sept. 1993, iv + 15 pp.  Second Revised Edition, May 1995.
  13. Sums of Powers of Numbers, 50 page manuscript, written as number theory class
    notes circa 1971 and later. Treats of Bernoulli, Stirling, Eulerian numbers.
  14. Tables of Combinatorial Identities and Series Techniques, Collated and edited by Dr. Jocelyn Quaintance, Based on the seven volumes of Gould's handwritten notes (1945-1990). Made Available on my WVU Departmental web site beginning 5 May 2010 as 8 PDF's which may be viewed or downloaded. This material is some 275 pages of formulas and identities.


MATHEMATICA MONONGALIAE

An Occasional Serial Founded by H. W. Gould in 1961

Issues 1-5 were originally issued as Technical Reports under auspices of an NSF
Research Grant. The name Mathematica Monongaliae was chosen to reflect the location
of West Virginia University in Monongalia County on  the Monongahela River.
The following numbers have been issued:

  1. H. W. Gould and A. T. Hopper, Generalized Hermite polynomials, 15 pp.
      - - Issued March 1961

  2. H. W. Gould, A binomial identity of Greenwood and Gleason, pp.1-6;
      H. W. Gould, A new convolution formula and some new orthogonal
          relations for inversion  of series, pp. 7-17;
      H. W. Gould, Congruences involving sums of binomial coefficients and a
          formula of Jensen, pp. 18-19;
      H. W. Gould, A note on range of summation, p.20.
       - - Issued August 1961

  3. H. W. Gould, Brill's series and some related expansions obtainable in
          closed form, 14 pp.
      - - Issued 16 Aug. 1961

  4. H. W. Gould, Exponential binomial coefficient series, 36 pp.
      - - Issued Sept. 1961

  5. H. W. Gould, Journal facilities for research in  mathematics at West
          Virginia University, 18 pp.
      - - Issued 1 Feb. 1962

  6. H. W. Gould, Notes on a calculus of Turán operators, 25 pp.
      - - Issued 18 May 1962
      N. B.:  First issue to carry the name 'Mathematica Monongaliae'

  7. W. Squire, Factorization of the operator, pp. 1-7;
      E. D. Casseday, Table of weights and abscissas for Gauss-Laguerre
          quadrature, pp. 8-121;
      H. W. Gould, Remarks on Fibonacci numbers, A bilinear generating
          function for Fibonacci numbers, pp. 13-20.
      - - Issued June 1962

  8. H. W. Gould, Generating functions for products of powers of Fibonacci and
          Lucas numbers, pp. 1-14;
      Problem Section, p.15;
      H. W. Gould, Note on recurrence relations for Stirling numbers, pp. 16-22.
      - - Issued July 1962

  9. H. W. Gould, Journal facilities for research in mathematics at West
          Virginia University, i +13 pp.
      - - Issued Oct. 1963

10. H. K. Fallin, Jr. and H. W. Gould, Chronological Bibliography of the Cauchy
          Integral Theorem,  25 pp.
      - - Issued May 1965;
      Revised edition, 32 pp. issued 1972, contains a list of 185 papers and
      books dealing with the theorem. A statistical chart is given, showing
      the number of items per decade. This monograph has been cited in
      Historia Mathematica, 1(1974), Nov., p. 508, as a definitive bibliography.

11. H. W. Gould, Journal facilities for research in mathematics at West
          Virginia University, pp. 3-17; Collected works of mathematicians
          available for study in the WVU libraries, pp. 18-20.
      - - Issued July 1968

12. H. W. Gould, Research bibliography of two special number sequences,
          Bell or exponential numbers, 135 items,pp.1-10; Euler-Fuss-Segner-
          Catalan numbers, 243 items, pp.11-25.  Introduction, pp. i-iv.
      - - Issued May 1971.
      Reviews in: Math. Reviews, 43(1972), #4755; Zentralblatt für Math.,
      226(1972), #10002; Math. of Computation, 26(1972), p. 302.
      Revised Edition issued 1976 listing 178 items on Bell and 444 items on
      Catalan numbers; viii + 38 pp., Issued by Combinatorial Research
      Institute, 1239 College Avenue, Morgantown, W. Va. 26505-5124
      Reviewed in Math. Reviews, 53(1977), #5460.
           Revised-corrected second printing, Jan.,1977. x + 39pp.
           Third printing, June 1977, with emendations. x + 41pp.
           Fifth printing, 26 Oct. 1979, with slight emendations. x + 43pp.
           Fifth Edition, 22 April 1985.  x + 43pp.
      (with Timothy J. Glatzer) New on-line PDF edition, 2011.

Mathematical Humor:

     The Nullset Review, a non-journal non-published irregularly and non-edited
           by H. W. Gould and Sam B. Nadler, Jr.  The Nullset Review is Devoted
           to the Study of Nothing at All. The editorial board also includes Non-editors
           A. A. Blank, M. T. Klass, I. M. Void, U. R. Null and other non-distinguished
           nobodies. Address: Gould at West Virginia University, or Prof. Nadler,
           Dept. of Math., University of Toledo, Toledo, Ohio 43606 (sam.nadler@utoledo.edu)

= = = = = = = =

Forthcoming:


The Bateman Notes
on
Binomial Coefficients



Edited and Revised by

Henry W. Gould



Department of Mathematics
West Virginia University
Morgantown, West Virginia


2010

= = = = = = = = = = = = = =

Dedicated to my Dear Wife Jean W. Gould
who remembers the intensity of my interest
 in the Bateman Notes in 1961, and
 because of her enthusiastic urging that now
 makes it possible for me to present
this work to the mathematical public.

= = = = = = = = = = = = = =

Preface


    This book is based on three manuscripts left by the late Professor Harry Bateman (1882-1946). Bateman taught at the California Institute of Technology. He had plans to produce some 25 books, but only three were published during his lifetime. This is all discussed in Volume 1 of the set of five books on special functions commonly referred to as the Bateman Manuscript Project that were published in his honor in 1953 after his  death. Cal Tech brought together four illustrious experts to edit Bateman's posthumous work. These experts were: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco Tricomi.

    The three manuscripts on binomial coefficients were loaned to the authur through the courtesy of the late Professor Arthur Erdélyi of the California Institute of Technology in 1960-61. This material was revised and edited in 1961 into the present typed and handwritten form. It had been hoped to put the material into better typed form or to use modern word processing techniques with a computer, but the author has been delayed for many years in this project, and it is hoped that the present Xerographic form will be of value until and if a better version may be produced.

    The Bateman Notes are quite incomplete and Bateman never produced a polished manuscript. Incomplete as the notes are, the author believes there is sufficient novelty in much of this material as to make it worthwhile to present the material to a wider mathematical audience having an interest in combinatorial identities. Indeed, a number of binomial identities in these Notes were included in the author's 1959 and 1972 book "Combinatorial Identities."

    Included in the present text are several instances where I have given proofs and remarks about some of the identities that interested Bateman.
                                                                                                   Morgantown, West Virginia,  6 August 2007

= = = = = = =

Attention is called to the collaboration of H. W. Gould
and Jocelyn Quaintance, WVU Visiting Research Assistant Professor.

The following joint research papers have been written since
August 2006:

1. A linear binomial recurrence and the Bell numbers and polynomials,
    Applicable Analysis and Discrete Mathematics,
    Vol. 1 (2007), No.2, pp. No. 2, 371-385  Available electronically at
    http: //pefmath.etf.bg.ac.yu   MR 2008g: #05014

2. Additional analysis on binomial recurrence coefficients.
    Applicable Analysis and Discrete Mathematics, Vol. 1 (2007), No. 2, 
    386-396.  Available electronically at
    http: //pefmath.etf.bg.ac.yu   MR 2008h:0 #05012

3. Floor and Roof Function Analogs of the Bell Numbers, Integers Journal,
    Vol. 7(1)(2007), Paper # A58, 21 pages.  MR 2008m: #11053
    Erratum published 10 Dec. 2009 as a PDF available with the paper on
    INTEGERS on Vol. 7 website.

4. Implications of Spivey's Bell number formula, 
    Journal of Integer Sequences, 11(2008), Issue 3, paper no. 7, 6pp.
    MR 2009h: #11033.

5. Products of numbers which obey a Fibonacci-type recurrence,
    Fibonacci Quarterly, 45(2007), Nov., No. 4, 337-346. Publ. Dec. 2008.
    MR 2009j: #11029.

6. Bell Numbers and Variant Sequences Derived from a General Functional 
    Differential Equation, Integers Journal, 9(2009), paper #A44, pp. 581-589.

7. Partial fraction expansions and a question of Bruckman,
    Fibonacci Quarterly, Vol. 46/47, No. 3, Aug., 2008/2009, pp. 245-248.
    MR 2010i: #05035.

8. Operators, inverse series relations and other expansions
    connected with generalized Gould-Hopper polynomials,
    Indian J. Math. 51(2009), No. 3, Dec. pp. 463-487.

9. Generalizations of Vosmansky's identity, Fibonacci Quarterly,
    Vol. 48(2010), No. 1, Feb. pp. 56-61.

10. Annihilation coefficients, binomial expansions and q-analogs, Journal of Mathematical
      Research and Exposition, Vol. 30(2010). No. 2, pp. 191-204.

11. Inverting a finite series with constant coefficients. Fibonacci Quarterly,
      Vol. 49(2011), No. 2, May, pp. 158-165.

12. A Ballot Number Formulary, Ars Combinatoria,  Vol. 102(2011),  365-383.  

13. A one parameter generalization of Bell's sum, Indian Journal of
      Mathematics, Vol. 53(2011), No 2, August, pp. 301-329.

14. Double fun with double factorials, Mathematics Magazine, 85(2012), No. 3, June, pp. 177-192. 

15. Bernoulli numbers and a new binomial transform identity, Journal of Integer Sequences, 17(2014), Issue  No..2,
      Article 14.2.2. 
      


Papers Submitted or in Process:

15. The Generalized Bell Number Recurrence and the Generalized
      Catalan Recurrence, being revised.

16. Sums of powers of binomial coefficients via Legendre
     polynomials, Part 3. being revised, April 2008.

17. Applications of Legendre polynomials to derive
      combinatorial identities, In preparation, March 2008.

18. A combinatorial identity of Walther Janous, March 2008.
      
19. Interpolating the double factorial function, March 2009.

20. A brief note on the differences of central binomial coefficients, March 2010.

21. New recurrences for Stirling numbers, July 2011.

23. Binomial identities of Ragnar Frish and Murray Klamkin, January 2012.
 

= = = = = = =

Gould Manuscript Project - Gould and Quaintance are preparing
a three-volume on-line tutorial and revision of Gould's manuscript Notes on Series.
It is expected that this project will require several years for completion and will be
web-accessible.The work is based on seven volumes (2000 handwritten pages) of
numbered results and identities which Gould put together from 1945 to 1990.

Now available: As a prelude Dr. Quaintance has prepared eight volumes of tables based on Gould's
seven volumes of unpublished notebooks on series. See the home page here for PDF
downloads.

Revised 26 Nov. 2014