On a new generalization of telephone numbers

Abstract

In this paper we introduce a new generalization of telephone numbers. We give the generating function, direct formulae, and matrix generators for these numbers. Moreover, we present their interpretations and we prove some properties of these numbers connected with congruences.

A note on telephone numbers and their matrix generators

“…Finally, based on [2] we found another generalized form that allows another interpretation for telephone numbers.…”

Certain Subclasses of Analytic Functions Associated with Generalized Telephone Numbers

“…In 2017, Wlochand Wolowiec-Musial [43] familiarized generalized telephone numbers (GTN) via the succeeding recursion T(κ, n) = κT(κ, n − 1) + (n − 1)T(κ, n − 2) n ≥ 0 and κ ≥ 1 with T(κ, 0) = 1, T(κ, 1) = κ, and deliberated some properties. In 2019, Bednarz and Wolowiec-Musial [2] presented a new generalization of TN by…”

“…T κ (3) = 1 + 3κ (4) T κ (4) = 1 + 6κ + 3κ 2 (5) T κ (5) = 1 + 10κ + 15κ 2 (6) T κ (6) = 1 + 15κ + 45κ 2 + 15κ 3 . In [7], forz ∈ ∆ := {z : z ∈ C and |z| < 1} an unit disc, Deniz consider…”