Transformation and summation formulas for double hypergeometric series

A number of new transformation formulas for double hypergeometric series are presented. The series appearing here are the so-called Kampé de Fériet functions of type F_1:1;2^0:3;4(1,1) and F_0:2;2^1:2;2(1,1). The transformation formulas relate such double series to a single hypergeometric series of ₄F₃(1) type. By specializing certain parameters, a list of new summation formulas for F_0:2;2^1:2;2(1,1) series is obtained. The origin of the results comes from studying symmetries of the 9-j coefficient appearing in quantum theory of angular momentum.     

Taylor series for the Askey-Wilson operator and classical summation formulas

An analogue of Taylor's formula, which arises by substituting the classical derivative by a divided difference operator of Askey-Wilson type, is developed here. We study the convergence of the associated Taylor series. Our results complement a recent work by Ismail and Stanton. Quite surprisingly, in some cases the Taylor polynomials converge to a function which differs from the original one. We provide explicit expressions for the integral remainder. As an application, we obtain some summation formulas for basic hypergeometric series. As far as we know, one of them is new. We conclude by studying the different forms of the binomial theorem in this context.

     

Hyperfunctions, formal groups and generalized Lipschitz summation formulas

A construction relating the theory of hyperfunctions with the theory of formal groups and generalizations of the classical Lipschitz summation formula is proposed. It involves new polylogarithmic rational functions constructed via the Fourier expansion of certain sequences of Bernoulli-type polynomials, related to the Lazard formal group. Related families of one-dimensional hyperfunctions are also constructed.     

Certain summation and transformation formulas for generalized hypergeometric series

We derive summation formulas for generalized hypergeometric series of unit argument, one of which upon specialization reduces to Minton’s summation theorem. As an application we deduce a reduction formula for a certain Kampé de Fériet function that in turn provides a Kummer-type transformation formula for the generalized hypergeometric function _pF_p(x).     

Two new trigonometric formulas with applications

We derive two new trigonometric identities and the corresponding Chebyshev identities. We also obtain new Fibonacci and Lucas Identities.     

Numerical integration by means of adapted Euler summation formulas

Summary The purpose of the paper is the study of formulas and methods for numerical integration based on Euler summation formulas. Cubature formulas are developed from multidimensional generalizations. Estimates of the truncation error are given in adaptation to specific properties of the integrand.     

Four families of summation formulas involving generalized harmonic numbers

In terms of Abel’s transformation on difference operators, we establish four families of summation formulas involving generalized harmonic numbers. They include several known and numerous new harmonic number identities as special cases.     

Spreading maps (polymorphisms), symmetries of poisson processes,and matching summation

The matrix of a permutation is a particular case of Markov transition matrices. In the same way, a measure-preserving bijection of a space (A, ) with a finite measure is a particular case of Markov transition operators. A Markov transition operator can also be considered as a map (polymorphism) (A, ) (A, ), which spreads points of (A, ) into measures on (A, ). Denote by * the multiplicative group of positive real numbers, and by the semigroup of measures on *. In this paper, we discuss *-polymorphisms and -polymorphisms, which are analogs of Markov transition operators (or polymorphisms) for the groups of bijections (A, ) (A, ) leaving the measure quasi-invariant; two types of polymorphisms correspond to the cases where A has finite and infinite measure, respectively. In the case where the space A itself is finite, the *-polymorphisms are some -valued matrices. We construct a functor from -polymorphisms to *-polymorphisms; it is described in terms of summations of -convolution products over matchings of Poisson configurations. Bibliography: 33 titles.Published in Zapiski Nauchnykh Seminarov POMI, Vol. 292, 2002, pp. 62–91.This revised version was published online in April 2005 with a corrected cover date and article title.     

Certain summation formulas involving harmonic numbers and generalized harmonic numbers

Harmonic numbers and generalized harmonic numbers have been studied since the distant past and involved in a wide range of diverse fields such as analysis of algorithms in computer science, various branches of number theory, elementary particle physics and theoretical physics. Here we aim at presenting further interesting identities about certain finite or infinite series involving harmonic numbers and generalized harmonic numbers by applying an algorithmic method to a known summation formula for the hypergeometric function ₅F₄(1).     

Character formulas and spectra of compact nilmanifolds

The authors give a new method for calculating the spectrum and multiplicities of the irreducible unitary representations appearing in the quasi-regular representation U: N × L²(ΓβN) → L²(ΓβN) on a compact nilmanifold ΓβN. They proceed by decomposing the trace of U into traces of irreducible representations. The basic calculations in the paper deal with lattice subgroups (Λ = log Γ an additive lattice in the Lie algebra $\text{N}$), essentially using the Poisson summation formula. Let Ad′ be the contragredient adjoint action of N on $\text{N}$¹. If ?₀ ? $\text{N}$¹, the multiplicity of π(?₀) in U is zero unless the Ad′(N) orbit of ?₀ meets $\text{Λ}{}^{\mathrm{\perp }}\text{= {h ? N}{}^1\text{: <h, Λ> ?}\text{Z}\text{}}$. If ?₀ ? Λ^⊥, then the multiplicity is a sum over representatives of certain Ad′(Γ)-orbits in,

$\text{m(π(?}{}_0\text{),U) =}\text{∑}\text{Ad′(N)?}{}_0\text{∩Λ}{}^{\mathrm{\perp }}\text{Ad′(Γ)}\text{k(?)}$

.The constants $\text{k(?)}$ are given both algebraic and geometric interpretations that lead to simple and effective calculations. Similar formulas hold if Γ is not a lattice subgroup.     

Short proofs of summation and transformation formulas for basic hypergeometric series

We show that several terminating summation and transformation formulas for basic hypergeometric series can be proved in a straightforward way. Along the same line, new finite forms of Jacobi's triple product identity and Watson's quintuple product identity are also proved.     

Recursive formulas for multinomial probabilities with applications

Recursive formulas are provided for computing probabilities of a multinomial distribution. Firstly, a recursive formula is provided for computing rectangular probabilities which include the cumulative distribution function as a special case. These rectangular probabilities can be used to provide goodness-of-fit tests for the cell probabilities. The probability that a certain cell count is the maximum of all cell counts is also considered, which can be used to assess the probability that the maximum cell count corresponds to the cell with the maximum probability. Finally, a recursive formula is provided for computing the probability that the cell counts satisfy a certain ordering, which can be used to assess the probability that the ordering of the cell counts corresponds to the ordering of the cell probabilities. The computational intensity of these recursive formulas is linear in the number of cells, and they provide opportunities for calculating probabilities that would otherwise be computationally challenging. Some examples of the applications of these formulas are provided.     

A generalization of the Lipschitz summation formula and some applications

The Lipschitz formula is extended to a two-variable form. While the theorem itself is of independent interest, we justify its existence further by indicating several applications in the theory of modular forms.