Combinatorial identities and hypergeometric functions. (English) Zbl 1506.05026

Alzer, Horst; Richards, Kendall C.

Rocky Mt. J. Math. 52, No. 6, 1921-1928 (2022).

Summary: We use properties of the Gaussian hypergeometric function to prove the following identities for combinatorial polynomials:
\[\sum_{j = 0}^n \binom{n + \alpha}{j} \binom{n+\beta}{n-j} z^j=\binom{n+\alpha}{n} \sum^n_{j=0}\binom{n}{j}\frac{\binom{n+j+\alpha+\beta}{j}}{\binom{j+\alpha}{j}} (z-1)^{n-j}\] and \[m\binom{m+n}{m} (1-z)^n \sum^n_{k=0} \frac{\binom{n}{k}}{m+k}\bigg(\frac{z}{1-z}\bigg)^k-\sum^n_{k=0}\binom{m+n}{k}(-z)^k=\sum^n_{k=0}\binom{m+n}{k}(-z)^{n-k}-\sum^n_{k=0}\binom{m+n}{n-k}(-z)^{n-k}.\] These formulas extend two combinatorial identities published by J. Brereton et al. [Electron. J. Comb. 18, No. 2, Research Paper P14, 14 p. (2011; Zbl 1229.05036)].

MSC:

05A19	Combinatorial identities, bijective combinatorics
33C05	Classical hypergeometric functions, \({}_2F_1\)
33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Keywords:

combinatorial identity; hypergeometric function; Jacobi polynomial

Citations:

Zbl 1229.05036

References:

[1]	H. Alzer, M. K. Kwong, and H. Pan, “Combinatorial identities and trigonometric inequalities”, Colloq. Math. 145:2 (2016), 291-305. · Zbl 1350.05009 · doi:10.4064/cm6859-5-2016
[2]	G. E. Andrews, R. Askey, and R. Roy, Special functions, Encyclopedia of Mathematics and its Applications 71, Cambridge University Press, 1999. · Zbl 0920.33001 · doi:10.1017/CBO9781107325937
[3]	R. Askey, Orthogonal polynomials and special functions, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1975. · Zbl 0298.33008
[4]	C. Banderier and S. Schwer, “Why Delannoy numbers?”, J. Statist. Plann. Inference 135:1 (2005), 40-54. · Zbl 1074.01012 · doi:10.1016/j.jspi.2005.02.004
[5]	J. Brereton, A. Farid, M. Karnib, G. Marple, A. Quenon, and A. Tefera, “Combinatorial and automated proofs of certain identities”, Electron. J. Combin. 18:2 (2011), art. id. 14. · Zbl 1229.05036 · doi:10.37236/2010
[6]	M. Petkovšek, H. S. Wilf, and D. Zeilberger, \[A=B\], A K Peters, Wellesley, MA, 1996. · Zbl 0848.05002
[7]	A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and series, vol. 3: More special functions, Gordon and Breach, New York, 1986 · Zbl 0733.00004