Evaluation of a Class of Binomial Coefficient Summations
نویسنده
چکیده
In this paper we prove the fo rmula ~o w ~-dk (Pn-bkk)(q +k bk) ( _ 1 ) k (p --kbW/d) (k q-kW/d) -I (p + q -k =kffio n k / and ob ta in a more general t r ans fo rma t ion for s u m s of the fo rm ~ (a +kbk) (c + b(n-k))f(k)g(n -k ). k=o n k These sums con ta in m a n y special cases in the l i terature of statistics, probabili ty, combinator ia l analysis, special funct ions , and o ther areas of ma themat ics .
منابع مشابه
Atomic three- and four-body recurrence formulas and related summations
A new recursive procedure is reported for the evaluation of certain three-body integrals involving exponentially correlated atomic orbitals. The procedure is more rapidly convergent than those reported earlier. The formulas are relevant to ab initio electronic-structure computations on threeand four-body systems. They also illustrate techniques that are useful in the evaluation of summations in...
متن کاملWeighted quadrature rules with binomial nodes
In this paper, a new class of a weighted quadrature rule is represented as -------------------------------------------- where is a weight function, are interpolation nodes, are the corresponding weight coefficients and denotes the error term. The general form of interpolation nodes are considered as that and we obtain the explicit expressions of the coefficients using the q-...
متن کاملOn Bivariate Generalized Exponential-Power Series Class of Distributions
In this paper, we introduce a new class of bivariate distributions by compounding the bivariate generalized exponential and power-series distributions. This new class contains the bivariate generalized exponential-Poisson, bivariate generalized exponential-logarithmic, bivariate generalized exponential-binomial and bivariate generalized exponential-negative binomial distributions as specia...
متن کاملSieved Partition Functions and Q-binomial Coefficients
Abstract. The q-binomial coefficient is a polynomial in q. Given an integer t and a residue class r modulo t, a sieved q-binomial coefficient is the sum of those terms whose exponents are congruent to r modulo t. In this paper explicit polynomial identities in q are given for sieved q-binomial coefficients. As a limiting case, generating functions for the sieved partition function are found as ...
متن کامل87. Binomial Coefficient Identities and Hypergeometric Series
In recent months I have come across many instances in which someone has found what they believe is a new result, in which they evaluate in closed form a sum involving binomial coefficients or factorials. In each case they have managed to do that either by using the recent powerful method of Wilf and Zeilberger (the W–Z method) [6], or by comparing coefficients in some ad hoc algebraic identity....
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2006