# Asymptotic formulae for mixed congruence stacks

Asymptotic formulae for mixed congruence stacks Much like the important work of Hardy and Ramanujan (Proc Lond Math Soc 2(17):75–115, 1919) proving the asymptotic formula for the partition function, Auluck (Math Proc Camb Philos Soc 47:679–686, 1951) and Wright (Quart J Math (Oxf) 22:107–116, 1971) gave similar formulas for unimodal sequences. Following the circle method of Wright, we provide the asymptotic expansion for unimodal sequences on a two-parameter family of mixed congruence relations, with parts on one side up to the peak satisfying $$r \pmod {m}$$ r ( mod m ) and parts on the other side $$-r\pmod {m}$$ - r ( mod m ) . Techniques used in the proofs include Wright’s circle method, modular transformations, and bounding of complex integrals. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Research in the Mathematical Sciences Springer Journals

# Asymptotic formulae for mixed congruence stacks

, Volume 5 (1) – Feb 6, 2018
14 pages

/lp/springer_journal/asymptotic-formulae-for-mixed-congruence-stacks-sV6A2Y0O1i
Publisher
Springer International Publishing
Subject
Mathematics; Mathematics, general; Applications of Mathematics; Computational Mathematics and Numerical Analysis
eISSN
2197-9847
D.O.I.
10.1007/s40687-018-0124-6
Publisher site
See Article on Publisher Site

### Abstract

Much like the important work of Hardy and Ramanujan (Proc Lond Math Soc 2(17):75–115, 1919) proving the asymptotic formula for the partition function, Auluck (Math Proc Camb Philos Soc 47:679–686, 1951) and Wright (Quart J Math (Oxf) 22:107–116, 1971) gave similar formulas for unimodal sequences. Following the circle method of Wright, we provide the asymptotic expansion for unimodal sequences on a two-parameter family of mixed congruence relations, with parts on one side up to the peak satisfying $$r \pmod {m}$$ r ( mod m ) and parts on the other side $$-r\pmod {m}$$ - r ( mod m ) . Techniques used in the proofs include Wright’s circle method, modular transformations, and bounding of complex integrals.

### Journal

Research in the Mathematical SciencesSpringer Journals

Published: Feb 6, 2018

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