REDUCE package for the indefinite and definite summation

Infinite derivative's series expansion of Indefinite and Definite Integral

In this article it is proven the existence of integration of indefinite integrals as infinite derivative's series expansion. This also opens a new way to integrate a definite integral.

Algorithms for the indefinite and definite summation

The celebrated Zeilberger algorithm which finds holonomic recurrence equations for definite sums of hypergeometric terms $F(n,k)$ is extended to certain nonhypergeometric terms. An expression $F(n,k)$ is called a hypergeometric term if both $F(n+1,k)/F(n,k)$ and $F(n,k+1)/F(n,k)$ are rational functions. Typical examples are ratios of products of exponentials, factorials, $Γ$ function terms, bin omial coefficients, and Pochhammer symbols that are integer-linear with respect to $n$ and $k$ in their arguments. We consider the more general case of ratios of products of exponentials, factorials, $Γ…

新的不定求和公式及其若干应用

New Indefinite Summation Formulas and Some Applications

本文提出了一种适用于任意给定函数的新型不定求和（反差分）公式，并将其应用于求解非齐次线性差分方程与不等式的特解，同时建立了该公式与现有结果之间的恒等关系。

重复主不定和

Repeated principal indefinite summation

本文研究了主不定和的重复迭代，建立了显式公式和收敛结果，并探讨了与多重伽马函数等特殊函数的联系。

A Novel Discriminant Approximation of Periodic Differential Equations

A new approximation of the discriminant of a second order periodic differential equation is presented as a recursive summation of the evaluation of its excitation function at different values of time. The new approximation is obtained, at first, by means of Walsh functions and then, by using some algebraic properties the dependence on the Walsh functions is eliminated. This new approximation is then used to calculate the boundaries of stability. We prove that by letting the summation elements number approach to infinite, the discriminant approximation can be rewritten as a summation of definit…

Summation, transformation, and expansion formulas for multibasic theta hypergeometric series

After reviewing some fundamental facts from the theory of theta hypergeometric series we derive, using indefinite summation, several summation, transformation, and expansion formulas for multibasic theta hypergeometric series. Some of the identities presented here generalize corresponding formulas given in Chapter 11 of the Gasper and Rahman book [Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics And Its Applications 96, Cambridge University Press, Cambridge, 2004].