Related papers: The generating function of the $\sigma_1$ function
Recently, Andrews, Dixit and Yee introduced partition functions associated with Ramanujan/Watson third order mock theta functions $\omega(q)$ and $\nu(q)$. In this paper, we find several new exact generating functions for those partition…
We discuss a method for computing the generating function for the multiplicity distribution in field theories with strong time dependent external sources. At leading order, the computation of the generating function reduces to finding a…
In this paper we study substitutions and some of their associated generating functions. This association takes aperiodicity to transcendence, and vice-versa. These generating functions have a recursive structure arising from the…
A representation of divisor function $\tau(n)\equiv \sigma_{0}(n)$ by means of logarithmic residue of a function of complex variable is suggested. This representation may be useful theoretical instrument for further investigations of…
The hypergeometric distribution is a popular distribution, whose properties have been extensively investigated. Generating functions of this distribution, such as the probability-generating function, the moment-generating function, and the…
We use the generating function of the characters of $C_2$ to obtain a generating function for the multiplicities of the weights entering in the irreducible representations of that simple Lie algebra. From this generating function we derive…
The study of Ramanujan-type congruences for functions specific to additive number theory has a long and rich history. Motivated by recent connections between divisor sums and overpartitions via congruences in arithmetic progressions, we…
In the sixth chapter of his notebooks Ramanujan introduced a method of summing divergent series which assigns to the series the value of the associated Euler-MacLaurin constant that arises by applying the Euler-MacLaurin summation formula…
We present new methods for the study of a class of generating functions introduced by the second author which carry some formal similarities with the Hurwitz zeta function. We prove functional identities which establish an explicit…
Let $T$ be an underlying space with a non-atomic measure $\sigma$ on it. In [{\it Comm.\ Math.\ Phys.}\ {\bf 292} (2009), 99--129] the Meixner class of non-commutative generalized stochastic processes with freely independent values,…
In this paper we attempt to prove Lehmer's conjecture on Ramanujan's tau function, namely tau(n) is never zero, for each n larger than zero by investigating the additive group structure attached to tau(n) with the aid of unique…
A solution is proposed for the problem of composition of ordinary generating functions. A new class of functions that provides a composition of ordinary generating functions is introduced; main theorems are presented; compositae are written…
The generating function method that we had developing has various applications in physics and not only interress undergraduate students but also physicists. We solve simply difficult problems or unsolved commonly used in quantum, nuclear…
We present an infinite family of identities that represent Ramanujan's tau function in terms of convolution sums of twisted divisor functions. Our method involves explicitly constructing non-vanishing level $1$ cusp forms from modular forms…
A generating function for a class of multigluon amplitudes is constructed as a particular solution of the self-duality equation.
Let $F(x)=\sum\limits_{n=1}^\infty\tau(n)x^n$ be the generating function for the number $\tau(n)$ of spanning trees in the circulant graphs $C_{n}(s_1,s_2,\ldots,s_k).$ We show that $F(x)$ is a rational function with integer coefficients…
In this paper we present a generating function approach to two counting problems in elementary quantum mechanics. The first is to find the total ways of distributing identical particles among different states. The second is to find the…
Let $\Bbb Z$ and $\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a_1,a_2,\ldots,a_k,n\in\Bbb N$ let $N(a_1,a_2,\ldots,a_k;n)$ be the number of representations of $n$ by…
Using some properties of the gamma function and the well-known Gauss summation formula for the classical hypergeometric series, we prove a four-parameter series expansion formula, which can produce infinitely many Ramanujan type series for…
In this paper we obtain asymptotic formulas for the Fourier coefficients of an infinite family of crank generating functions. Moreover we use this result to show that the crank obeys certain inequalities. This implies that the crank can not…