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).}

- Note on a paper of Grosswald, Amer. Math. Monthly, 65(1954),
251-253.

MR 15(1954), p.685. - Some generalizations of Vandermonde's convolution, Amer.
Math.

Monthly, 63(1956), 84-91. MR 17(1956), p. 702. - Final analysis of Vandermonde's convolution, Amer. Math.
Monthly,

64(1957), 409-415. MR 19(1958), p.379. - A theorem concerning the Bernstein polynomials, Math. Mag.,
31(1958),

259-264. MR 20(1959), #5997. - Note on a paper of Sparre Andersen, Math. Scand., 6(1958),
226-230.

MR 21(1960), p.491. - Dixon's series expressed as a convolution, Nordisk Mat. Tidsk.,
7(1959),

73-76. Summary on p. 96. MR 22(1961), #115 - Note on a paper of Steinberg, Math. Mag., 33(1959), 46-48.

MR 21(1960), #4924 - An iterative approximation for finding the n-th root of a
number,

Math. Mag., 33(1959), 61-69. MR 23(1962) #B556. - Generalizations of a theorem of Jensen concerning convolutions,
Duke

Math. J., 27(1960), 71-76. MR 22(1961), #5585. - Stirling number representation problems, Proc. Amer. Math. Soc.,

11(1960), 447-451. MR 22(1961), #5586. - The Lagrange interpolation formula and Stirling numbers,
Proc. Amer.

Math. Soc., 11(1960), 421-425. MR 22(1961), #5588. - A q-binomial coefficient series transform, Bull. Amer. Math.
Soc., 66

(1960), 383-387. MR 22(1961), #9764. - Fjeldstad's series and the q-analog of a series of Dougall, Math.
Scand.,

8(1960), 198-200. MR 23(1962), #A1546. - The q-series generalization of a formula of Sparre Andersen,
Math.

Scand., 9(1961), 90-94. MR 23(1962), #A810. - Note on a Paper of Klamkin concerning Stirling numbers, Amer.
Math.

Monthly, 68(1961), 477-479. MR 23(1962), #A1548. - A series transformation for finding convolution identities, Duke
Math. J.,

28(1961), 193-202. MR 23(1962) #A1216. - The q-Stirling numbers of first and second kinds,. Duke Math. J.,

28(1961), 281-289. MR 23(1962), #A99. - Some relations involving the finite harmonic series, Math. Mag.,

34(1961), 317-321. MR 25(1963), #2000. - A generalization of a problem of L. Lorch and L. Moser, Canadian
Math.

Bull., 4(1961), 303-305. - A binomial identity of Greenwood and Gleason, The Mathematics
Student,

29(1961), 53-57. MR 29(1965), #5754. - (with A. T. Hopper) Operational formulas connected with two

generalizations of Hermite polynomials, Duke Math. J., 29(1962),

51-63. MR 24(1962), #A2689. - (with Remy Landau), Floating primes, Recreational Math.
Magazine,

Issue No.8, April 1962, 34-35. - Congruences involving sums of binomial coefficients and a formula
of

Jensen, Amer. Math. Monthly, 69(1962), 400-402.

MR 38(1969), #2033. - (with T. A. Chapman) Some curious limits involving definite
integrals,

Amer. Math. Monthly, 69(1962), 651-653. ZBL 108(1964), 270-

271. - A new convolution formula and some new orthogonal relations
for

inversion of series, Duke Math. J., 29(1962), 393-404. MR 25(1963),

#3333. - Generalizations of an integral formula of Bateman, Duke Math. J.,

29(1962), 475-479. MR 25(1963), #3201. - Theory of binomial sums, Proc. W. Va. Acad. Sci., 34(1962), 158-

162(1963). MR 36(1968), #3660. - (with William Squire) Maclaurin's second formula and its
generalization,

Amer. Math. Monthly, 70(1963), 44-52. MR 26(1963), #4073. - Operational Recurrences involving Fibonacci numbers, Fibonacci
Quart.,

1(1963), No.1, 30-33. MR 27(1964), #1402. - Note on two binomial coefficient sums found by Riordan, Annals of
Math.

Stat., 34(1963), 333-335. MR 26(1963), #47. - The q-series analogue of a formula of Szily, Scripta Mathematica,
26(1961),

155-157 (1963). MR 26(1963), #6073. - Generating functions for products of powers of Fibonacci and
Lucas

numbers, Fibonacci Quart., 1(1963) , No.2, 1-16. MR 33(1967), #93. - Generalization of a bracket function formula of L. Moser,
Canadian

Math. Bull. 6(1963), 275-277. ZBL 113(1965), p. 267. - Isotone and antitone fractions, Math. Magazine, 36(1963), 240-
244.

ZBL 116(1965), p. 41; RZ 1964, #7b3. - A general Túran expression for the Zeta function,
Canadian
Math. Bull.

6(1963), 359-366. MR 28(1964), #2099. - Fibonacci exponentials and generalizations of Hermite
polynomials,

Fibonacci Quart., 1(1963), No.4, 31-34. ZBL 147(1968), p. 21. - Sums of logarithms of binomial coefficients, Amer. Math.
Monthly,

71(1964), 55-58. ZBL 123(1966), p. 2. RZ 1964), #10A134. - 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. - (with L. Carlitz) Bracket function congruences for binomial
coefficients,

Math. Mag., 37(1964), 91-93. ZBL 128(1967), p. 41. - A new series transform with applications to Bessel,
Legendre, and

Tchebycheff polynomials, Duke Math. J., 31(1964), 325-334.

MR 28(1964), #4272. - Distance of closest approach of an artificial earth satellite,
Proc. W. Va.

Acad. Sci., 36(1964), 169-174 (1964). - The operator (a^x)Delta)^n and Stirling numbers of the
first kind, Amer. Math.

Monthly, 71(1964), 850-858. MR 29(1965), #4709. - 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) - The construction of orthogonal and quasi-orthogonal number
sets,

Amer. Math. Monthly, 72(1965), 591-602. MR 31(1966), #2161. - An identity involving Stirling numbers, Annals of Inst. of
Statistical

Math. (Tokyo), 17(1965), 265-269. MR 31(1966), #5808. - Non-Fibonacci numbers, Fibonacci Quart., 3(1965), 177-183.

MR 35(1968), #118. - A variant of Pascal's triangle, Fibonacci Quart., 3(1965),
257-271.

Corrections of misprints, ibid., 4(1966), 62. MR 36(1968), #5006. - Inverse series relations and other expansions involving Humbert

polynomials, Duke Math. J., 32(1965), 697-711. MR 32(1966), #5948. - Cale's rule for planetary distances, Proc. W. Va. Acad. Sci.,
37(1965),

243-257 (1966). - Note on recurrence relations for Stirling numbers, Publ. l'Inst.
Math.

Belgrade, Nouv.Sér. 6(20)(1966), 115-119. MR 34(1967), #4190. - (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. - Note on a q-series transform, J. Combinatorial Theory, 1(1966),
397-399.

MR 34(1967), #1745. - (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. - A bibliography of research in mathematics at West Virginia
University,

1867-1967, Proc. W. Va. Acad. Sci., 38(1966), 123-133 (1967). - Note on a combinatorial identity in the theory of bi-colored
graphs,

Fibonacci Quart., 5(1967), 247-250. MR 36(1968), #5024. - History of the old telescope and observatory at West Virginia
University,

Proc. W. Va. Acad. Sci., 39(1967), 1-8 (1968). - 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. - Some remarks on backward pairs, Mathematics Notes, Washington
State

Univ., 11(1968), No. 4, Nov., 1-2. - Note on a paper of Paul Byrd, and a solution of problem P-3,
Fibonacci

Quart., 6(1968), 318-321. MR 39(1970), #4465. - Note on two binomial coefficient identities of Rosenbaum, J.
Math.

Physics, 10(1969), 49-50. ZBL 179(1970), p. 318. RZ 1969, #8B159. - 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. - Power sum identities for arbitrary symmetric arrays, SIAM J.
Applied

Math., 17(1969), 307-316. MR 40(1970), #52. - Invariance and other simple techniques as a genesis of
binomial

identities, Nieuw Archief voor Wiskunde, (3)17(1969), 214-217.

MR 41(1971), #78. - (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). - 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. - Remarks on a paper of Srivastava, J. Mathematical Physics,
12(1971),

1378-1379. MR 44(1972), #2622. - Comment on a paper of Chiang, J. Mathematical Physics, 12(1971), 1743.
- 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. - Equal products of generalized binomial coefficients, Fibonacci
Quart.,

9(1971), 337-346. MR 45(1973), #6641. - Explicit formulas for Bernoulli numbers, Amer. Math. Monthly,
79(1972),

44-51. MR 46(1973), #5229. - (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 - A new symmetrical combinatorial identity, J. Combinatorial
Theory,

Ser. A, 13(1972), 278-286. MR 46(1973), #3328. - 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. - 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. - 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. - 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. - A triangle with integral sides and area, Fibonacci Quart.,
11(1973),

27-39. Errata, ibid., 11(1973), 294. MR 46(1973), #8970. - A remarkable combinatorial formula of Heselden, Proc. W. Va.
Acad. Sci.,

43(1971), 198-201 (1973). MR 50(1975), #4327. - (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. - 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. - Some combinatorial identities: Comment on a paper of
Srivastava,

Nordisk Matematisk Tidskrift, 21(1973), 7-9. MR 48(1974), #3746. - 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. - Coefficient identities for powers of Taylor and Dirichlet series,
Amer.

Math. Monthly, 81(1974), 3-14. MR 49(1975), #4795. - An expansion of the operator (x^(p+q)Delta^n f(x), Glasnik
Matematicki,

(3)8(28)(1973), 259-272. MR 48(1974), #6736. - (with L. C. Hsu) Some new inverse series relations, Duke Math.
J.,

40(1973), 885-891. MR 49(1975), #2421. - 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. - Note on functions studied by LeVan, Chawla, and Euler, J. Natur.
Sci. & Math.,

Lahore, Pakistan, 12(1972), 421-430 (1974). MR 50(1975), #218. - (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). - 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. - The design of the four binomial identities: Moriarty
intervenes,

Fibonacci Quart., 12(1974), 300-308. MR 50(1975), #9609. - 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. - Some sequences generated by spiral sieving methods, Fibonacci
Quart.,

12(1974), 393-397. ZBL 305(1976), #10006. RZ 1975, #7A163. - Formal proof of equivalence of two solutions of the general
Pascal

recurrence, Fibonacci Quart., 13(1975), 127-128. MR 50(1975), #9772. - Remarks on some formulas of Trumper, Fibonacci Quart.,
14(1976),

143, 146. Published as a Letter to the Editor. MR 54(1977), #2577. - Types of orthogonality for arrays of numbers, Proc. W. Va. Acad.
Sci.,

47(1975), 219-228 (1977). ZBL 365(1978), #15013. - A Fibonacci formula of Lucas and its subsequent manifestations
and

rediscoveries, Fibonacci Quart., 15(1977), 25-29. MR 55(1978), #126. - A rearrangement of series based on a partition of the natural
numbers,

Fibonacci Quart., 15(1977), 67-72. MR 55(1978), #2740. - Estimation of actuarial functions arising in coal mine
valuation,

Proc. W. Va. Acad. Sci., 49(1977), 67-83. ZBL 366(1978), #40001. - Generalization of a formula of Touchard for Catalan numbers,

J. Combinatorial Theory, Ser. A, 23(1977), 351-353. MR 56(1978), #8376. - (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. - 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) - Generalization of a formula of Hilden and Fenger, Nordisk
Matematisk

Tidskrift, 24(1976), 109-110 (1977). Summary on p. 135. MR 58(1979),

#255. - 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. - Evaluation of sums of convolved powers using Stirling and
Eulerian

numbers, Fibonacci Quart., 16(1978), 488-497, 560-561. MR 80b: #05007. - Operational formulas for unusual Fibonacci series, Fibonacci
Quart.,

16(1978), 555-560. MR 80b: #05008. - A history of the Fibonacci Q-matrix and a higher-dimensional
problem,

Fibonacci Quart., 19(1981), 250-257. MR 82j: #01055. - (with M. E. Mays) Series expansions of means, J. Math. Anal. and
Appl.,

101(1984), 611-621. MR 86a: #26025. - (with W. E. Greig) A Lucas triangle primality criterion dual to
that of

Mann-Shanks, Fibonacci Quart., 23(1985), 66-69. MR 86h: #11007. - New inverse series relations for finite and infinite series with

applications, J. Math. Research and Exposition, 4(1984), 119-130.

MR 87k: #40003. - (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. - The g-inverse and reverse-converse series inverse, Indian J. Pure
and

Appl. Math., 17(1986), 613-628. MR 87m: #05019. - Number-theoretic identities involving the Möbius function
and the

logarithm, Indian J. Math., Ramanujan Centenary Vol., 29(1987),107-115.

MR 89d; #11003. - (with M. E. Mays) Counting chains of numbers, Utilitas
Mathematica,

31(1987), 227-232. MR 88h: #05006. - Some formulas for Euler's constant, Bull. Number Theory and
Related

Topics, Buenos Aires, 10(1986), No.2, 2-9 (1988). MR 89g: #11011. - (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. - 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. - (with H. Gingold and M. E. Mays) Power product expansions,
Utilitas

Mathematica, 34(1988), 143-161. MR 90i: #40003. - Series transformations for finding recurrences for
sequences, Fibonacci

Quarterly, 28(1990), No. 2, May, 166-171. MR 91b: #11016. - A bracket function series transform and its inverse, Fibonacci
Quarterly,

32(1994), No. 2, May, 176-179. MR 95e: #11031. - A class of binomial sums and a series transform, Utilitas
Mathematica,

45(1994), May, 71-83. MR 95c: #05006. - (with Paula Schlesinger) Extensions of the Hermite g. c. d.
theorems

for binomial coefficients, Fibonacci Quarterly, 33(1995), 386-391.

MR 97g: #11015 - 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. - (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. - (with Temba Shonhiwa) Functions of GCD's and LCM's, Indian
Journal

of Math. (Allahabad), 39(1997), No.1, 11-36. MR 98j: #11001 - (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 - 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.

- 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

- (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. - The generalized chain rule of differentiation with historical
notes,

Utilitas Mathematica, 61(2002), May, 97-106. MR 2003a:#26007 - 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)

- 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. - Proof and generalization of a Catalan number formula of
Larcombe,

Congressus Numerantium, 165(2003), 33-38. MR 2004m: #05023. - A curious identity which is not so curious. Mathematical
Gazette

(London, England 88(2004), Issue No. 511, March, p. 87. - Sums and convolved sums of Catalan numbers and their
generating

functions, Indian Journal of Math. 46(2004), Nos. 2-3, 137-160. - Sums of powers of binomial coefficients via Legendre
polynomials,

Ars Combinatoria, 73(2004), 33-43. MR 2006a: #11019. - (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)

- Remarks on Compositions of Numbers into Relatively Prime Parts,

Notes on Number Theory and Discrete Mathematics, 11(2005), No. 3. pp.1-6. - Classes of iterated binomial transforms, Indian J. Math., 47(2005), No.2-3,

pp.183-203. MR 2007i: #39033. - A baker's dozen proofs of a binomial identity, Publikacije Elektrotehnickog

Fakulteta, Ser. Mat., 16(2005), 131-138. MR 2006c: #05025. - Curious special function identities. (A generalization of No. 131 above)

Publikacije Elektrotehnickog Fakulteta (Univ. Beograd), Ser. Mat., 18(2006),

22-27. MR 2008b: #05016 - The inverse of a finite series and a third-order recurrent sequence,

Fibonacci Quarterly, 44(2006), Nov. 302-315. MR 2008f: #11016. - (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. - Higher order extensions of Melzak's formula, Utilitas
Mathematica,

Vol. 72(2007), 23-32. MR 2008b: #05009. - Note on a combinatorial formula of Rosenberg, Bulletin of the Institute

of Combinatorics and Its Applications, Vol. 51(2007), Sept. pp. 5-12.

- (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 - (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 - (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. - Sums of powers of binomial coefficients via Legendre polynomials, Part 2,

Ars Combinatoria, Vol. 86(2008), Jan. 161-173. MR 2008m: #33022. - A note on Combinatorial Identities arising from the Lagrange-Waring

Interpolation Formula, Ars Combinatoria, Vol, 86(2008), 281-288. MR 2008m: #05029. - (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. - (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.

- (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. - Proof and generalization of an Identity of E. T. Bell, Bulletin of the Allahabad Math.

Society, 24(2009), Part 1, pp. 71-82, - (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. - (with Jocelyn Quaintance) Bell Numbers and Variant Sequences Derived from a

General Functional Differential Equation, Integers Journal, 9(2009), paper #A44, pp. 581-589. - (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.

- Kai Lai Chung (1917-2009): In Memoriam, Journal of Mathematical Research and Exposition,

(Dalian, China) Vol. 29(2009), No. 6, pp. 952-953. - (with Jocelyn Quaintance) Generalizations of Vosmansky's identity, Fibonacci Quarterly,

Vol. 48(2010), No. 1, Feb. pp. 56-61. - (with Jocelyn Quaintance) Annihilation coefficients, binomial expansions and q-analogs, Journal

of Mathematical Research and Exposition, Vol. 30(2010). No. 2, pp. 191-204.

- (with Salar Y. Alsardary) Special function inverse series pairs, Kyungpook Mathematical Journal,

Vol. 50(2010), No. 2, pp. 155-193. - (with Jocelyn Quaintance) Inverting a finite series with constant coefficients, Fibonacci Quarterly, 49(2011), May, No. 158-165
- (with Timothy J. Glatzer) Catalan and Bell Numbers, Research Bibliography,

New on-line edition 2011. Download PDF here: BibGood final.pdf - (with Jocelyn Quaintance) A Ballot Number Formulary, Ars Combinatoria,

Vol. 102(2011), 365-383. - (with Jocelyn Quaintance) A one parameter generalization of Bell's sum, Indian Journal of

Mathematics, Vol.53(2011), No. 2, August, pp. 301 -329. - (with Jocelyn Quaintance) Double fun with double factorials, Mathematics Magazine, 85(2012), No. 3, June, pp. 177-192.
- (with Joel V. Brawley and John Brillhart) Recollections of Leonard Carlitz, Acta Arithmetica, 152(2012), 361-372.
- (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)
- (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.
- (with Jocelyn Quaintance) Bernoulli numbers and a new binomial transform identity, Journal of Integer Sequences, 17(2014), Issue No..2, Article 14.2.2.
- 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.
- (with Harris Kwong and Jocelyn Quaintance) On certain sums of Stirling numbers

with binomial coefficients, Journal of Integer Sequences. 18(2015), Article 15.9.6

- 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.
- 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.
- Book Review of: 1962 Chelsea reprint of 'Introduction to the Operations with Series' by I. J. Schwatt; Amer. Math. Monthly, 70(1963), 107.
- Book Review of: 'Operational Calculus and Generalized Functions' by A. Erdélyi, N.Y., 1962; Amer. Math. Monthly, 71(1964), 574.
- 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.
- 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.
- Letter to the Editor, Notices of Amer. Math. Soc., 16(1969), No. 4, June, p. 624. Concerns upside-down Aleph font in Russian journals.
- Book Review of: 'A History of π (Pi)', by Petr Beckman, Sec. Ed., Boulder, Colorado, 1971; Math. of Computation, 28(1974), 325-327.
- 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.
- 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.
- Book Review of: 'Die mathematischen Studien von G. W. Leibniz zur Kombinatorik' by Eberhard Knobloch, Wiesbaden, 1973; ISIS 67(1976), 315-316.
- Book Review of: 'Die mathematischen Studien von G. W . Leibniz zur Kombinatorik. Textband' by Eberhard Knobloch, Wiesbaden, 1976; ISIS 71(1980), 178.
- Book Review of: 'Der Beginn der Determinantentheorie: Leibnizens nachgelassene Studien zum Determinantenkalkul. Textband', by Eberhard Knobloch, Hildesheim, 1980; ISIS, 73(1982), 119.
- Which numbers are interesting?, The Mathematics Teacher, 73(1980), 408. Letter to the Editor.
- LAFF GRAFFS, A Contribution to Mathematical Humor and Wit, published

by the author, Morgantown, 1987. i + 15pp. - A critical view: The new editor looks at journal standards, Proc. W. Va. Acad. Sci., 46(1974),

No.1, xviii-xxi (1975). Editorial. - Outgoing editor's closing remarks, Proc. W. Va. Acad. Sci., 50(1978), pp. x-xiii (1980).
- 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.
- Solution of Prob. 4211, Amer. Math. Monthly, 63(1956), 126-127, Ref. to paper of mine, pp. 84-91.
- Solution of Prob. 375, Math. Mag. 33(1959), 115-117. A sum posed by D. A. Steinberg.
- Problem E 1388, Amer. Math. Monthly, 66(1959), 813. Soln. by N.J. Fine, ibid. 67(1960), 476.
- Solution of Prob. 470, Math. Mag. 36(1963), 206. Posed by W. Squire.
- Solution of Prob. 505, Math. Mag., 35(1963), 267-269. Posed by Carlitz.
- Solution of Prob. 521, Math. Mag., 37(1964), 61. Posed by L. Moser.
- (with T. A. Chapman) Solution of functional equations involving Túran expessions,

Proc. W. Va. Acad. Sci., 35(1963), 205(1964). Abstract. - Properties of polynomials associated with expansions of certain difference operators,

Proc. W. Va. Acad. Sci., 35(1963), 208(1964). Abstract. - Solution of Problem 548, Math. Mag., 37(1964), 357-358. Posed by L. Moser.
- Problem 5231, Amer. Math. Monthly, 71(1964), 923. Solns by Klamkin, Carlitz, and Riordan,

ibid., 72(1965), 921-923. - Associativity and the golden mean, Fibonacci Quarterly, 2(1964), 203.
- Prob. E 1767, Amer. Math. Monthly, 72(1965), 315. Soln by E. S. Langford,

ibid., 73(1966), 540. - 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. - Solution of Prob. 571, Math. Mag., 38(1965), 185-187.

- A Fibonacci crossword puzzle, Fibonacci Quart., 4(1966), 59-61; Soln, ibid., p.150.
- Comment on Prob. 602, Math. Mag., 40(1967), 107-108. Posed by Bruce W. King.
- (with Maurice Machover) Problem E 1975, Amer. Math. Monthly, 74(1967), 437-438.

Solution by M. T. L. Bizley, ibid. 75(1968), 682-683. - Solution of Prob. 67-5, SIAM Review, 10(1968), 225-226. Posed by D. J. Newman

W. Weissblum.

- Late comment on Prob. 67-14, SIAM Review, 10(1968), 453. (inverse of Newton-Girard

formula) - (with Franklin C. Smith) Problem 5612, Amer. Math. Monthly, 75(1968), 791.

Soln by M. T. L. Bizley, ibid., 76(1969), 705-707. - Solution of Prob. 5541, Amer. Math. Monthly, 72(1968), 1019-1021. Posed by Robert Breusch. Ramanujan's exponential limit.
- Comment on Prob. 612, Math. Mag., 39(1966), No. 4, 253. Joint comment with William Squire.

- Solution of Prob. 676, Math. Mag., 41(1968), 165-166. Posed by William Squire.
- Solution of Prob. 5463, Amer. Math. Monthly, 75(1968), 204-205. Posed W. G. Spohn.
- Problem H-142, Fibonacci Quart., 6(1968), 252. Soln by L. Carlitz, ibid. 8(1970), 275-276.
- Power sum identities for arbitrary symmetric arrays, Proc. W. Va. Acad. Sci., 40(1968),

364 (1969). Abstract. - Problem 5777*, Amer. Math. Monthly, 78(1971), 202. Extension of Hermite's g.c.d. theorems

for binomial coefficients. - Solution of Prob. 783, Math. Mag., 44(1971), 289-291, Posed by H. L. Krall.
- Solution of Prob. 70-21, SIAM Review, 14(1972), 166-169. Posed by V. K. Rohatgi.
- Problem 825, Math. Mag., 45(1972), 100-101. Solns by V. Konecny and C. W. Trigg,

ibid., 46(1973), 45-46. - Comment on a paper of Wiggins on an occupancy problem, Math. Mag., 45(1972), in a paragraph
- 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.
- 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.
- Solution of Prob. 72-22, SIAM Review, 15(1973), 798-799. Posed by Barry Wolk.
- (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.
- Soln. of Prob. H-219, Fibonacci Quart., 13(1975), 185-187. Posed by Paul Bruckman. (Solution generalizes a binomial identity)
- Solution of Prob. 974, Math. Mag., 50(1977), 216. Posed by John P. Hoyt.
- Remark on Prob. H-123, Fibonacci Quart., 16(1978), 189. Posed by Douglas Lind.
- 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.
- About Fibonacci Numbers, posted as a Facebook Note, Feb. 2011.
- Bell and Catalan Numbers, posted as a Facebook Note, 20 March 2011.

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.

- A formula for sums of powers of numbers, written in 1945, to be revised.
- Sums and convolved sums of Rothe coefficients,
- Worpitzky's formula for Bernoulli numbers and generalizations,
- Differences of the harmonic series, Stirling numbers, and

q-analogs, Being revised. - (with J. Wetweerapong) q-analogs of some binomial identities.
- Generalized binomial theorem and identities,
- Persymmetric matrices and determinants of Catalan numbers,
- Persymmetric matrices and determinants of Bell numbers,
- A new inverse series relation motivated by the Bell numbers,
- A Short Catalog of q-Binomial Coefficient Identities.
- (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. - (with A. M. Chak) Expansions of Multidimensional Catalan Numbers,
- 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, - Historical note on the four-color conjecture, the work of C. N.

Reynolds, and remarks about libraries and mathematics departments,

(written 1992, revised 1997) - Determinants of multifactorials.
- The coefficients in the series expansion of y = (1+1/x)^x.
- Novel combinatorial identities analogous to those of Vandermonde, Part 1,
- Novel combinatorial identities analogous to those of Vandermonde, Part 2,
- Howard Eves and other great mathematical catalysts who made all things possible,
- Some binomial identities of Munarini,
- Three combinatorial identities of Seiffert,
- An integration trick, (pedagogical note)
- An identity of Oliver Lung-Feng Chen,
- 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.) - 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, - An extension of the Truesdell F-equation, long memoir,
- Fibonomial Catalan numbers: Arithmetic properties and a table of the

first fifty numbers, - A Catalogue of 100 Postulate Sets for Boolean Algebra, (Begun c. 1954

in collaboration with the late Henry Warren Thwing at U.Va.), - Proofs of some theorems in Boolean algebras. written circa 1954,
- 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), - (with J. B. Kim) On matrix semigroups and a q-function related to

theta functions, - (with Hollie L. Buchanan) Legal numbers, a problem in permutations,
- Some L. C. M. properties of the binomial coefficients, in collaboration

with Paul Erdös, and assisted by Arsen Melkumian, - Some new inverse series relations and new formulas for Stirling

numbers, Cf. Notices of Amer. Math. Soc., 21(1974), p. A-603, - 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. - 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), - (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. - Stirling number, binomial coefficient, and summation notations,

and some personal reminiscences of mathematics, 1992; 2003 - 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. - Remarks concerning Euler's constant and a collection of 100 formulas

for this number, - Generalized Bernoulli and Euler polynomial convolution identities.

written Dec. 1994, revised 2001. - Riordan's proof of Grosswald's identity,
- Some Combinatorial identities of Brodzki,
- Some trees in my forest,
- Mathematical Identities, An Investigation into the Nature of Research,
- History of Combinatorial Identities.
- The Magic Eleven and the Lucas Team Identities, joint endeavor with Mike Long,
- Alice in Boola-Boola-Land, with apologies to Lewis Carroll,

- Bibliography on the chain rule of differentation,
- Combinatorial identities derived from powers of 2 x 2 matrices,
- The origin, evolution and naming of two sciences and their practitioners,
- The emergence of combinatorics as a separate mathematical discipline,

and the parallel case of behaviorology versus psychology. - Notes on Fibonacci numbers, written in May 1979, expository.
- (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). - The origin and application of combinatorial identities, Benedum

Distinguished Scholar Award lecture, 29 March 1988. Revised 31 Jan.

2004. - A series identity and Legendre polynomials of positive and negative

index, - The interchange of order of summation in double sums,
- Operations with series and formal power series: A tutorial paper, 1993
- The interchange of order of summation in double sums, Nov. 1962.
- Notes on reciprocal derivatives, begun in 1948.
- A quadratic recurrence, paper begun with George E. Trapp, c. 1970's.
- A formula of Hayes for sums of logarithms of binomial coefficients,
- Lagrange interpolation and generalization of a theorem of Howard,
- (with the late Wilbur. H. Bluhm) "Fractionons", Fraction-sensitive means,

original version sent to College Math. Journal; being revised. - Factoring quadratics, to submit to a math. education magazine.
- (with the late Vadim Komkov) The log of a sum and symmetrical function

expansions, - (with T. A. Chapman) A finite Laplace transform,
- 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) - How to wrap a package: Some notes on geodesics,

(Have a cardboard model of these geodesics) - A Euclidean geometric dissection problem,
- Remarks on the equation f'(ax) + f(x) = 0,

- Length of a spiral of uniform pitch,
- A congruence of Gegenbauer for number-theoretic sums,
- A bibliography on convolution relations and related series,

A five-page list giving 58 references. Compiled Dec. 1957 at U.N.C. - 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. - 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. - How to choose a calculus book . . . by weight!, 25 pp. statistical tables, 1986.

In collaboration with the late Prof. Wilbur H. Bluhm. - Chromatic polynomials, 13 April 1987; revised 8 Dec. 1992; revised

28 Oct. 1992. - Some terminology suggested by matrix theory (linear algebra),

24 Jan. 1991; Revised 15 Jan. 1993. - Proof as convergence,
- A ternary Farey process,
- Generalized trigonometry,
- 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. - (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. - Baldwin's 'trisection', a classroom note, written 27 - 29 March 1997.
- Two geometric constructions, trisection of an angle and a more

general figure, Quillen's trisection method, - (with John H. Biggs) Comments on Presentation in Sigley and Stratton's

'Solid Geometry', written 10 October 1960. (Math. 7 supplement) - Mathematical Identities, An Investigation into the Nature of Research,

Monograph notes for Math. 407, Fall 1997, WVU. 36 pp. - A mathematical formula for the binding energy of all the isotopes

of atomic elements 1 through 118. Formula developed c. 1945.

unpublished. - 572957795130823208767981548141 . . . AND ALL THAT!

written 24 June 2005. Pedagogical note. Answers query from Bruce Welker. - The atomic origins of my scientific curiosity, written 10 Jan. 2006

and posted on my WVU Mathematics Department web site under

'U235 Discovery.html'. - Karl Friedrich Gauss and Benjamin Apthorp Gould, 9 July 2007. revised.
- Word Unprocessing, a new phenomenon, written 16 March 2009.

posted on my WVU Math. web site. - (with Timothy J. Glatzer) Further classes of iterated binomial transforms

and related bracket function transforms, submitted. - Limits of sums involving logarithms of binomial coefficients, 6 June 2006.
- Complex number extensions of combinatorial identities as a source of

new combinatorial identities, given 13 Oct. 1993, Seminar, WVU - A Bernoulli number identity of Carlitz and a new binomial transform,
- Generalization of symmetrical binomial identities of Redl and Catalan,
- Generalized hypergeometric form of the Rothe expansions.
- (with Jocelyn Quaintance) A brief note on sums whose general term is

a convolution, Under revision. - Note on a paper of Porter and identities of Abel type, written

27 February 2007. Being revised. - (with Jocelyn Quaintance) The generalized Bell number recurrence

and Catalan recurrence, 12 July 2007. - (with Jocelyn Quaintance) Research Proposal: Catalan-Like Recurrences

With Arbitrary Initial Conditions, 21 Jan. 2007. - (with Jocelyn Quaintance) Sums of powers of binomial coefficients via Legendre

polynomials, Part 3. February 2008, being revised. - (with Jocelyn Quaintance) Applications of Legendre polynomials to derive

combinatorial identities, In preparation, March 2008. - (with Jocelyn Quaintance) A combinatorial identity of Walther Janous, March 2008.
- (with Jocelyn Quaintance) Interpolating the double factorial function, March 2009.
- (with Jocelyn Quaintance) A brief note on the differences of central binomial coefficients,

March 2010. - (with Jocelyn Quaintance) New recurrences for Stirling numbers, July 2011.
- (with Jocelyn Quaintance) A Bernoulli Number Identity Posed by Carlitz and a

New Binomial Transform of Chen, written 15 Dec. 2011. Submitted, 2012 - (with Jocelyn Quaintance) Binomial identities of Ragnar Frish and Murray Klamkin,

January 2012. Submitted. - Novel Combinatorial Identities Analogous to those of Vandermonde, Part 1. Submitted to

SCIENCE CHINA MATHEMATICS special issue in honor of Prof. Leetsch C.

Hsu's 95th birthday in 2015.(To appear. in Chinese) 与Vandermonde恒等式类似的新奇组合恒等式：第一部分

H. W. Gould, Books - Monographs

- Combinatorial Identities, First edition. U. S. Department of Defense, Nov. 1959.
- 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. - Topics in Combinatorics, published by the author, Second
edition, Lecture Notes.

April 2000. iii+120pp. - Catalan and Bell Numbers: Research Bibliography of Two
Special

Number Sequences, fifth edition, 1979. (= Mathematica Monongaliae No. 12.) - (with Timothy J. Glatzer) Catalan and Bell Numbers, Research

Bibliography, New on-line edition 2011. Download PDF here: BibGood final.pdf

- 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. - 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. - The Bateman Notes on Binomial Coefficients, book written in 1961, to be published.
- Handbook of Postulate Sets for Boolean Algebra, In preparation.
- 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. - 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. - Sums of Powers of Numbers, 50 page manuscript, written as number theory class

notes circa 1971 and later. Treats of Bernoulli, Stirling, Eulerian numbers. - 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.

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.

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 (

= = = = = = = =

Forthcoming:

The Bateman Notes

on

Binomial Coefficients

on

Binomial Coefficients

Edited and Revised by

Henry W. Gould

Henry W. Gould

Department of Mathematics

West Virginia University

Morgantown, West Virginia

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.

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

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

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

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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.

16. (with Harris Kwong and Jocelyn Quaintance) On certain sums of Stirling numbers

with binomial coefficients, 25 June 2015, Journal of Integer equences. Vol. 18 (2015), Article 15.9.6

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.

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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 20 August 2015