On b-ary binomial coefficients with negative entries

TL;DR

This paper extends the concept of b-ary binomial coefficients to include negative entries, exploring their properties, explicit formulas, and generalizations that satisfy Pascal-like recurrences.

Contribution

It introduces a generalization of b-ary binomial coefficients with negative entries, along with new properties, explicit formulas, and multiple generalizations.

Findings

Derived explicit expressions involving restricted partitions.

Established symmetry, congruence, and Pascal-like recurrence properties.

Proposed two generalizations partially satisfying Pascal-like recurrences.

Abstract

Abstract. We generalize the $b$-ary binomial coefficients with negative entries, which is based on the generating function obtained in early work. Besides an explicit expression involving the restricted partition, several properties such as symmetry, congruence and Pascal-like recurrence are studies. Finally, we also provide two different generalizations, partially satisfying Pascal-like recurrences.