5 papers
Two complementary relations for the Rogers-Ramanujan continued fraction
Sumit Kumar Jha
Let be the Rogers-Ramanujan continued fraction. We give different proofs of two complementary relations for given by Ramanujan and proved by Watson and Ramanathan. Ou…
An identity for
Sumit Kumar Jha
We prove that $$ \prod_{n=0}^{\infty}(1+q^{10n+5}) = \frac{\sum_{n=-\infty}^{\infty}q^{n^{2}}\, \sum_{n=-\infty}^{\infty}(-1)^{n}\, (-q)^{n(3n-1)/2}}{4\, \sum_{n\geq 0}(-q)^{\frac{…
An identity involving number of representations of as a sum of triangular numbers
Sumit Kumar Jha
Let denote sum over divisors of a positive integer , and denote the number of representations of as a sum of triangular numbers. Then we prove th…
Formulas for the number of -colored partitions and the number of plane partitions of in terms of the Bell polynomials
Sumit Kumar Jha
We derive closed formulas for the number of -coloured partitions and the number of plane partitions of in terms of the Bell polynomials.
A formula for the -coloured partition function in terms of the sum of divisors function and its inverse
Sumit Kumar Jha
Let denote the -coloured partition function, and denote the sum of positive divisors of . The aim of this note is to prove the following $$ p_{…