Reciprocal series are employed to systematically review convolution sums, orthogonality relations, recurrence relations and reciprocal formulae for several classical number sequences, such as binomial coefficients, Stirling numbers, Bernoulli Euler numbers.

Explicit formulas for solutions of recurrence relations for 3–loop vacuum integrals are generalized for the n-loop case. Recently [1] a new approach to implement recurrence relations [2] for vacuum integrals was suggested. Such relations connect Feynman integrals with various degrees of their denominators. In many cases they provide a possibility to express an integral with given degrees of den...

We study Poincaré recurrence from a purely geometrical viewpoint. In [8] it was proven that the metric entropy is given by the exponential growth rate of return times to dynamical balls. Here we use combinatorial arguments to provide an alternative and more direct proof of this result and to prove that minimal return times to dynamical balls grow linearly with respect to its length. Some relati...

We establish two recurrence relations for some Clausen’s hypergeometric functions with unit argument. solve them to give the explicit formulas. Additionally, we use moments of Ramanujan’s generalized elliptic integrals obtain these relations.

We present a computer algebra approach to proving identities on Bernoulli and Euler polynomials by using the extended Zeilberger’s algorithm given by Chen, Hou and Mu. The key idea is to use the contour integral definitions of the Bernoulli and Euler numbers to establish recurrence relations on the integrands. Such recurrence relations have certain parameter free properties which lead to the re...

The aim of this paper is to define and study of the Gegenbauer matrix polynomials of two variables. An explicit representation, a three-term matrix recurrence relations, differential recurrence relations and hypergeometric matrix representation for the Gegenbauer matrix polynomials of two variables are given. The Gegenbauer matrix polynomials are solutions of the matrix differential equations a...

Shannon, A.G., J.C. Turner and K.T. Atanassov, A generalized tableau associated with colored convolution trees, Discrete Mathematics 92 (1991) 329-349. This paper considers the tableaux arising from the colors at different levels of arbitrary order convolution trees. The results are related to the elements which constitute the Pascal-LucasTurner triangles and to various generalizations of the F...

In this paper, we provide some recurrence relations connecting restricted partition functions and mock theta functions. Elementary manipulations are used including Jacobi triple product identity, Euler’s pentagonal number theorem, Ramanujan’s for finding the relations.