Hyper-Dual $q$-Francois Numbers

Abstract

In this paper, we introduce and study a new class of hyper-dual numbers, called hyper-dual $q$-Francois numbers, by combining the concepts of $q$-integers, Francois numbers, and hyper-dual numbers. The Binet-type formula is derived for the proposed numbers. Moreover, both the ordinary generating function and the exponential generating function associated with the hyper-dual $q$-Francois numbers are obtained. Several summation formulas are also presented. In addition, Vajda-type identity for hyper-dual $q$-Francois numbers is proved. As special cases of this identity, well-known identities such as the Catalan-type, Cassini-type, and d’Ocagne-type identities are derived.

Keywords

• Francois numbers
• Hyper-dual numbers
• $q$-integers

References

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