Narrow Results

This paper presents the basic scheme and the log-normal, log-gamma and log-negative-inverted-gamma scenarios to establish the Rényi function for infinite products of homogeneous isotropic random fields on Rⁿ; in particular for random fields on the sphere in R³. The motivation of this paper is the test of (non-)Gaussianity on the Cosmic Microwave Background Radiation (CMBR) in Cosmology. In the presentation, we need to employ spherical harmonics for some concrete computations.

In this short paper, we demonstrate the computation of certain Glaisher-type products over the primes by differentiation of the Euler product identity. We are able to give a formula for the logarithmic derivative of ζ(x) in terms of this product, in addition to some other interesting identities.

Ramanujan's formula for the number r₂₄(n) of representations of a positive integer n as a sum of 24 squares asserts that

Let $q$ denote a complex variable with $\left|q\right|<1$. For a positive integer $k$ let

This paper examines the phenomenon of coefficients that vanish in a class of infinite products related to those first examined by Hirschhorn, and later by Tang, Baruah and Kaur, and the first author of this paper. Several infinite families of vanishing coefficient phenomena are found.

In particular, it is shown that if p is a prime in one of several arithmetic progressions modulo 24, then there exist integer values for j, v and w such that if b is any integer (or in same cases, any odd integer, or any even integer), then a result of the following type holds:

If the sequence $\{r_n\}$ is defined by

For $\left|q\right|<1$, define $f_i={\prod }_{n=1}^{\infty }(1-q^{in})$, and let $(A(q),B(q))$ be any of the pairs $(f_1^4,\frac{f_1^8}{f_2^2}),$$(f_1^4,\frac{f_1^{10}}{f_3^2}),$$(f_1^6,\frac{f_2^4}{f_1^2}),$$(f_1^6,\frac{f_1^{14}}{f_2^4}),$$(f_1^{10},\frac{f_2^6}{f_1^2}),$$(f_1^{14},\frac{f_3^5}{f_1}),(f_1^{14},\frac{f_2^8}{f_1^2}).$ For any such pair $(A(q),B(q))$, define the sequences $\{a(n)\}$ and $\{b(n)\}$ to be the coefficients of $q^n$ of $A(q)$ and $B(q)$, respectively. Then for each pair it is shown that $a(n)$ vanishes if and only if $b(n)$ vanishes. In each case, a criterion is given which states precisely when $a(n)=b(n)=0$. Moreover, for the pairs $(f_1^{26},\frac{f_3^9}{f_1}),$$(f_1^{26},\frac{f_2^{16}}{f_1^6})$ it is shown that $a(n)=b(n)=0$ if $12n+13$ satisfies a criteria of Serre for $a(n)=0$.

Andrews and Bressoud, Alladi and Gordon, and others, have proven, in a number of papers, that the coefficients in various arithmetic progressions in the series expansions of certain infinite q-products vanish. In the present paper, it is shown that these results follow automatically (simply by specializing parameters) in an identity derived from a special case of Ramanujan’s ₁Ψ₁ identity.

Likewise, a number of authors have proven results about the m-dissections of certain infinite q-products using various methods. It is shown that many of these m-dissections also follow automatically (again simply by specializing parameters) from this same identity alluded to above.

Two identities that may be considered as extensions of two identities of Ramanujan are also derived.

It is also shown how applying similar ideas to certain other Lambert series gives rise to some rather curious q-series identities, such as, for any positive integer m,

Applications to the Fine function F (a, b; t) are also considered.