In this article, we provide partition-theoretic interpretations for some new truncated pentagonal number theorem and identities of Gauss. Also, we deduce few inequalities for some partition functions.
In this article, we provide partition-theoretic interpretations for some new truncated pentagonal number theorem and identities of Gauss. Also, we deduce few inequalities for some partition functions.
Abstract In this work, we investigate the arithmetic properties of $b_{5^k}(n)$ , which counts the partitions of n where no part is divisible by $5^k$ . By constructing generating functions …
Abstract In this work, we investigate the arithmetic properties of $b_{5^k}(n)$ , which counts the partitions of n where no part is divisible by $5^k$ . By constructing generating functions for $b_{5^k}(n)$ across specific arithmetic progressions, we establish a set of Ramanujan-type congruences.
Abstract In this work, we investigate the arithmetic properties of $b_{5^k}(n)$ , which counts the partitions of n where no part is divisible by $5^k$ . By constructing generating functions …
Abstract In this work, we investigate the arithmetic properties of $b_{5^k}(n)$ , which counts the partitions of n where no part is divisible by $5^k$ . By constructing generating functions for $b_{5^k}(n)$ across specific arithmetic progressions, we establish a set of Ramanujan-type congruences.
In this article, we provide partition-theoretic interpretations for some new truncated pentagonal number theorem and identities of Gauss. Also, we deduce few inequalities for some partition functions.
In this article, we provide partition-theoretic interpretations for some new truncated pentagonal number theorem and identities of Gauss. Also, we deduce few inequalities for some partition functions.
Ramanujan conjecture for powers of 5 Abstract.Ramanujan conjectured, and G. N. Watson proved, that if n is of a specific form then p(n), the number of partitions of n, is …
Ramanujan conjecture for powers of 5 Abstract.Ramanujan conjectured, and G. N. Watson proved, that if n is of a specific form then p(n), the number of partitions of n, is divisible by a high power of 5.In the present note, we establish appropriate generating function formulae, from which the truth of Ramanujan's conjecture, as well as some results of a similar type due to Watson, are shown to follow easily.Furthermore, we derive two new congruences for the partition function.Our proofs are more straightforward than those of Watson and more recent writers and use only classical identities of Euler and Jacobi.