Untrodden pathways in the theory of the restricted partition function p(n,N)

Dixit, Atul; Eyyunni, Pramod; Maji, Bibekananda; Sood, G.

doi:10.1016/j.jcta.2021.105423

Cited by 6 publications

(8 citation statements)

References 31 publications

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“…and gave its partition theoretic interpretation [14,Corollary 4.3]. As a special case of Theorem 1.2, we are able to obtain a new series presentation for σ(q, N):…”

Section: Introductionmentioning

confidence: 91%

On Sum-Of-Tails Identities

Gupta

2020

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In this article, a finite analogue of the generalized sum-of-tails identity of Andrews and Freitas is obtained. We derive several interesting results as special cases of this analogue, in particular, a recent identity of Dixit, Eyyyunni, Maji and Sood. We derive a new extension of Abel's lemma with the help of which we obtain a one-parameter generalization of a sum-of-tails identity of Andrews, Garvan and Liang, an identity of Ramanujan as well as two new results -one for Ramanujan's function σ(q) and another for the function recently introduced by Andrews and Ballantine. Later we introduce a new generalization FFW c (n) of a function of Fokkink, Fokkink and Wang and derive an identity for its generating function. This gives, as a special case, a recent representation for the generating function of spt(n) given by Andrews, Garvan and Liang. We also obtain some weighted partition identities along with new representations for two of Ramanujan's third order mock theta functions through combinatorial techniques.

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“…and gave its partition theoretic interpretation [14,Corollary 4.3]. As a special case of Theorem 1.2, we are able to obtain a new series presentation for σ(q, N):…”

Section: Introductionmentioning

confidence: 91%

On Sum-Of-Tails Identities

Gupta

2020

Preprint

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“…It was shown in [14] that Theorem 1.2 has many fruitful consequences in the theory of partitions including a representation for spt(n, N), the number of smallest parts in all partitions of n whose corresponding largest parts are less than or equal to N, in terms of the restricted partition function p(n, N), which counts the number of partitions of n whose largest parts do not exceed N, and a finite analogue of the second Atkin-Garvan rank moment. See Theorem 2.4 of [14]. The work in [14] further extended the sparsely developed theory of p(n, N).…”

Section: Introductionmentioning

confidence: 92%

“…In a recent work [14], Eyyunni, Maji, Sood and the first author obtained a finite analogue of Theorem 1.1 given below.…”

Section: Introductionmentioning

confidence: 94%

“…See Theorem 2.4 of [14]. The work in [14] further extended the sparsely developed theory of p(n, N). Note that lim N→∞ p(n, N) = p(n), the number of unrestricted partitions of a positive integer n.…”

Section: Introductionmentioning

confidence: 95%

“…The inner sum is now handled using Heine's transformation [15, p. 359, Equations (III.1), (III.2)], namely, for |z| < 1 and We also need the following lemma whose proof is similar to that of Corollary 4.1 of [14]. Hence we give only the outline of the proof.…”

Section: An Identity Involving a Finite Sum Of A 2 φmentioning

confidence: 99%

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A finite analogue of a $q$-series identity of Bhoria, Eyyunni and Maji and its applications

Dixit¹,

Patel²

2022

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Bhoria, Eyyunni and Maji recently obtained a four-parameter q-series identity which gives as special cases not only all five entries of Ramanujan on pages 354 and 355 of his second notebook but also allows them to obtain an analytical proof of a result of Bressoud and Subbarao. Here, we obtain a finite analogue of their identity which naturally gives finite analogues of Ramanujan's results. Using one of these finite analogues, we deduce an identity for a finite sum involving a 2 φ 1 . This identity is then applied to obtain a generalization of the generating function version of Andrews' famous identity for the smallest parts function spt(n). The q-series which generalizes ∑ ∞ n=1 spt(n)q n is completely different from S(z, q) considered by Andrews, Garvan and Liang. Further applications of our identity are given. Lastly we generalize a result of Andrews, Chan and Kim which involves the first odd moments of rank and crank.

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