H. W. Gould

Some Generalizations of Vandermonde's Convolution

(1956). Some Generalizations of Vandermondes Convolution. The American Mathematical Monthly: Vol. 63, No. 2, pp. 84-91.

Coefficient Identities for Powers of Taylor and Dirichlet Series

giving a way to calculate as many of the Bs as desired. Formula (1.2), stated in tha way or in the form (1.3) or in various other forms, has been known a long time. Thus, in 1748 Euler gave (1.3) (obscurely expressed) in section 76 of his Introductio [4], [5]. Adams and Hippisley [1] gave the formula in the form (1.2) (their formula 6.361). Formula (1.3) is given as formula (0.314) in any of the several editions (Russian original, German, or English) of Ryshik and Gradstein [16], where, however, p is restricted to be a natural number. Thinking to remove this restriction of Ryshik and Gradstein, von Holdt [17] published still another derivation using properties of double sums to establish that (1.2) holds true for rational real p. His derivation avoids use of differentiation of the series in (1.1). We mention this because differentiation of formal power series affords a quick proof of (1.2) and has been used before. The basic recurrence relation is not as widely known as it should be, and has been rediscovered repeatedly. Thus Barrucand [2] found (1.2) again and made applications of it. Cappellucci [18] found it in the form (1.3) but attributed it to Hansted [19]. Hindenburg [12], in his treatise on the multinomial theorem (p. 291) gave (1.3) in a perfectly obscure notation. Many other references could be cited. Actually, the recurrence is implicit in still another class of formulas widespread in the literature, stemming from the early work of Hindenburgs student Rothe [15]. I myself have written a number of papers, e.g., [6]-[11] having to do with a formula of Rothe and its consequences for combinatorics, special functions and number theory. What we shall do here is to derive the formulas again and put them in a

Evaluation of a class of binomial coefficient summations

Abstract In this paper we prove the formula ∑ k = 0 n w w + d k ( p − b k n − k ) ( q − b k k ) = ∑ k = 0 n ( − 1 ) k ( p − b w / d k ) ( k + w / d k ) − 1 ( p + q − k n − k ) and obtain a more general transformation for sums of the form. ∑ k = 0 n ( a + b k k ) ( c + b ( n − k ) n − k ) f ( k ) g ( n − k ) These sums contain many special cases in the literature of statistics, probability, combinatorial analysis, special functions, and other areas of mathematics.

Series expansions of means

Abstract A mean M(u, v) is defined to be a homogeneous symmetric function of two positive real variables satisfying min(u, v) ⩽ M(u, v) ⩽ max(u, v) for all u and v. Setting M(u, v) = uM(1, vu−1) = uM(1, 1 − t), 0 ⩽ t μ p (1, 1 − t) = [ 1 2 + (1 − t) p 2 ] 1 p , m p (u, v) = [ (v p + 1 − u p + 1 ) (v − u)(p + 1) ] 1 p (Stolarskys mean), M p (u, v) = (u p + v p ) (u p− 1 + v p − 1 ) (Lehmers mean), E(r, s; u, v) = [ r(u s − v s ) s(u r − v r ) ] 1 (s − r) (Leach and Sholanders mean), and G(r, s; u, v) = [ (u s + v s ) (u r + v r ) ] 1 (s − r) (Ginis mean). The explicit power series coefficients and recurrence relations for these coefficients are found. Finally, applications are shown by proving a theorem that generalizes one due to Lehmer.

Combinatorial identities for Stirling numbers : the unpublished notes of H.W. Gould

This book is a unique work which provides an in-depth exploration into the mathematical expertise, philosophy, and knowledge of H W Gould. It is written in a style that is accessible to the reader with basic mathematical knowledge, and yet contains material that will be of interest to the specialist in enumerative combinatorics. This book begins with exposition on the combinatorial and algebraic techniques that Professor Gould uses for proving binomial identities. These techniques are then applied to develop formulas which relate Stirling numbers of the second kind to Stirling numbers of the first kind. Professor Goulds techniques also provide connections between both types of Stirling numbers and Bernoulli numbers. Professor Gould believes his research success comes from his intuition on how to discover combinatorial identities.This book will appeal to a wide audience and may be used either as lecture notes for a beginning graduate level combinatorics class, or as a research supplement for the specialist in enumerative combinatorics.

A new symmetrical combinatorial ldentity

Abstract A proof is given for the novel identity [ x+a a ][ y+b b ]=∑ k=0 m [ x+y+k k ][ x+a−b a−k ][ y+b−a b−k ] q (a−k)(b−k) due to Richard Stanley, where m = min(a, b), and brackets denote q-binomial coefficients defined by [ x n ] = Φ j=1 n q x−j+1 −1 q j −1 , [ x 0 ] = 1 The identity is valid for all real or complex x, y and non-negative integers a, b. The case q = 1 is shown to follow from a non-symmetrical identity of Nanjundiah, while the general case follows from the q-from of the Vandermonde convolution. A variant form of Stanleys identity is also given.

Double Fun with Double Factorials

Summary The double factorial of n may be defined inductively by (n + 2)!! = (n + 2)(n)!! with (0)!! = (1)!! = 1. Alternatively we may define this notion by the two relations (2n)!! = 2 ·4 · 6 · 8…(2n) =2nn! and (2n - 1)!! = 1 · 3 · 5 · 7…(2n - 1) = (2n)!/2n!. Our object is to exhibit some properties and identities for the double factorials. Furthermore, we extend the notion of double factorial to the binomial coefficients by introducing double factorial binomial coefficients. The double factorial binomial coefficient is defined as We derive identities and generating functions involving these double factorial binomial coefficients.

Note on Two Binomial Coefficient Identities of Rosenbaum

This paper gives rapid proofs of two binomial coefficient identities found by Rosenbaum [J. Math. Phys. 8, 1977 (1967)] who obtained the identities from rather involved considerations of commutation relations. The present proofs make use of the Vandermonde convolution, or addition, theorem and a well‐known fact that the kth difference of a polynomial of degree k − 1 is zero. In a sense the two special cases are not essentially new.

Generalization of a formula of Touchard for Catalan numbers

Abstract In this paper a general identity is proved which includes among its special cases a recurrence formula for Catalan numbers given by J. Touchard in 1924.

Bell Numbers and Variant Sequences Derived from a General Functional Differential Equation

Abstract It is well known that the Bell numbers have exponential generating function , which satisfies the differential equation . In this paper, we investigate certain sequences whose exponential generating functions satisfy a modified form of the above differential equation, namely, the functional differential equation . For the main result of this paper, we show that when a = –1 and b ∈ ℝ, the sequence obeys the simple second-order linear recurrence G(n + 2) = bG(n + 1) – G(n). The proof is based on a well-known binomial series inversion formula.