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An Alternative Generating Function for $k$-Regular Partitions

Kağan Kurşungöz

Abstract

We construct a $k$-fold $q$-series as a generating function of $k$-regular partitions for each positive integer $k$. The $k=1$ case is one of Euler's $q$-series identities pertaining to the partitions into distinct parts. The construction is combinatorial. Although we find a connection to Bessel polynomials in the $k=2$ case, this note is certainly not a study of Bessel polynomials and their $q$-analogs.

An Alternative Generating Function for $k$-Regular Partitions

Abstract

We construct a -fold -series as a generating function of -regular partitions for each positive integer . The case is one of Euler's -series identities pertaining to the partitions into distinct parts. The construction is combinatorial. Although we find a connection to Bessel polynomials in the case, this note is certainly not a study of Bessel polynomials and their -analogs.

Paper Structure

This paper contains 3 theorems, 41 equations.

Key Result

Theorem 1

where $b(0,0) = 1$, $b(m, n) = 0$ if $m<0$ or $n<0$, and

Theorems & Definitions (6)

  • Theorem 1
  • Lemma 2
  • proof
  • proof : proof of Theorem \ref{['thm2RegMain']}
  • Theorem 3
  • Conjecture 4