Some Combinatorial and Analytical Identities

Abstract

We give new proofs and explain the origin of several combinatorial identities of Fu and Lascoux, Dilcher, Prodinger, Uchimura, and Chen and Liu. We use the theory of basic hypergeometric functions, and generalize these identities. We also exploit the theory of polynomial expansions in the Wilson and Askey-Wilson bases to derive new identities which are not in the hierarchy of basic hypergeometric series. We demonstrate that a Lagrange interpolation formula always leads to very-well-poised basic hypergeometric series. As applications we prove that the Watson transformation of a balanced 4 φ 3 to a very-well-poised 8 φ 7 is equivalent to the Rodrigues-type formula for the Askey-Wilson polynomials. By applying the Leibniz formula for the Askey-Wilson operator we also establish the 8 φ 7 summation theorem.

Combinatorial identities from contour integrals of rational functions

“…It has been recently pointed out in [13] that the Ismail-Stanton result (Theorem 2.2) in [15] is also the special case of the above formula. However, it is worth noticing that by using the theory of basic hypergeometric functions the origin of several combinatorial identities of Fu and Lascoux, Dilcher, Prodinger, Uchimura, and Chen and Liu was well explained in [15]. Here we presented a different proof using contour integrals which is direct and simple.…”

“…It is a common generalization of Dilcher's identity [7] and of some identities due to Fu and Lascoux [9,10]. It has been recently pointed out in [13] that the Ismail-Stanton result (Theorem 2.2) in [15] is also the special case of the above formula. However, it is worth noticing that by using the theory of basic hypergeometric functions the origin of several combinatorial identities of Fu and Lascoux, Dilcher, Prodinger, Uchimura, and Chen and Liu was well explained in [15].…”

“…Recently, there has been an interest in q-identities related to divisor functions; see for example [7,9,10,[13][14][15]18,19,26,27] and references therein. Various methods such as divided difference, hypergeometric series, combinatorial proof and partial fraction decomposition have been proposed.…”

Coefficients of multiplication formulas for classical orthogonal polynomials

“…The operator D x is appropriate for B n (a, x), whereas the corresponding operator for the basis {ϑ n (a, x)} is the Wilson operator ( [9,20]) defined for an even function f by (a, x)) n , the multiplication formula…”

Some q-congruences for homogeneous and quasi-homogeneous multiple q-harmonic sums

“…With this motivation, several authors have recently investigated (28) and they extended it along several directions: see for example [1,3,9,10,12,13,20,22,24,25,31,32]. In [23], Prodinger shows the inversion of (28),…”