Some elementary problems on infinite products

viXra, 2018

I derive some Ser's infinite product for exponential function and exponential of the digamma function; as well as an integral representation for the digamma function.

Journal of Mathematical Analysis and Applications, 1991

Gulf Journal of Mathematics

In this paper, we give the relationship between ℐ-convergent infinite product and ℐ-convergent infinite series. We also give some results related to these concepts.

Journal of the Australian Mathematical Society, 2014

We extend results of Andrews and Bressoud [‘Vanishing coefficients in infinite product expansions’, J. Aust. Math. Soc. Ser. A27(2) (1979), 199–202] on the vanishing of coefficients in the series expansions of certain infinite products. These results have the form that if $$\begin{eqnarray}\frac{(q^{r-tk},q^{mk-(r-tk)};q^{mk})_{\infty }}{(\pm q^{r},\pm q^{mk-r};q^{mk})_{\infty }}=:\mathop{\sum }_{n=0}^{\infty }c_{n}q^{n}\end{eqnarray}$$ for certain integers $k$, $m$, $s$ and $t$, where $r=sm+t$, then $c_{kn-rs}$ is always zero. Our theorems also partly give a simpler reformulation of results of Alladi and Gordon [‘Vanishing coefficients in the expansion of products of Rogers–Ramanujan type’, in: The Rademacher Legacy to Mathematics (University Park, PA, 1992), Contemporary Mathematics, 166 (American Mathematical Society, Providence, RI, 1994), 129–139], but also give results for cases not covered by the theorems of Alladi and Gordon. We also give some interpretations of the analytic...

viXra, 2018

Journal of the Australian Mathematical Society, 1979

Richmond and Szekeres (1977) have conjecturned that certain of the coefficients in the power series expansions of certain infinite products vanish. In this paper, we prove a general family of results of this nature which includes the above conjectures.

Journal of Applied Mathematics and Computing, 2013

We introduce a family of discrete analytic functions, called expandable discrete analytic functions, which includes discrete analytic polynomials, and define two products in this family. The first one is defined in a way similar to the Cauchy-Kovalevskaya product of hyperholomorphic functions, and allows us to define rational discrete analytic functions. To define the second product we need a new space of entire functions which is contractively included in the Fock space. We study in this space some counterparts of Schur analysis.

Canadian Journal of Mathematics, 1995

Let ƒ(z) be a complex function analytic in some neighbourhood of the origin with ƒ(0) = 1. It is known that ƒ(z) admits a unique "power product" expansion of the form convergent near zero. We derive a simple direct bound for the radius of convergence of this product expansion in terms of the coefficients of ƒ(z). In addition we show that the same bound holds in the case of "inverse power product" expansions Examples are given for which these bounds are sharp. We show also that products with nonnegative coefficients have the same radius of convergence as their corresponding series.

An innovatory approach has been recently proposed for the derivation of infinite product representation of elementary functions. The approach is based on the comparison of different alternative forms of Green's functions for boundary-value problems stated for the twodimensional Laplace equation. A number of new infinite product representations of elementary functions was actually derived within the scope of that approach. The present study continues the trend: it aims at an analysis of the approach and exploring ways for its extending to some other problem statements that might also be efficiently treated.

Journal of Approximation Theory, 1989