The aim of this paper is to introduce several degenerate hyperbolic functions as degenerate versions of the hyperbolic functions, to evaluate Volkenborn and the fermionic $p$-adic integrals of the degenerate hyperbolic cosine and the degenerate hyperbolic sine functions and to derive from them some identities involving the degenerate Bernoulli numbers, the degenerate Euler numbers and the Cauchy numbers of the first kind.

Bell polynomials are widely applied in many problems arising from physics and engineering. The aim of this paper is to introduce new types of special polynomials and numbers, namely Bell polynomials and numbers of the second kind and poly-Bell polynomials and numbers of the second kind, and to derive their explicit expressions, recurrence relations and some identities involving those polynomials and numbers. We also consider degenerate versions of those polynomials and numbers, namely degenerate Bell polynomials and numbers of the second kind and degenerate poly-Bell polynomials and numbers of the second kind, and deduce their similar results.

In this paper, we derive some identities and recurrence relations for the degenerate Stirling numbers of the first kind and of the second kind, which are degenerate versions of the ordinary Stirling numbers of the first kind and of the second kind. They are deduced from the normal orderings of degenerate integral powers of the number operator and their inversions, certain relations of boson operators and the recurrence relations of the Stirling numbers themselves. Here we note that, while the normal ordering of an integral power of the number operator is expressed with the help of the Stirling numbers of the second kind, that of a degenerate integral power of the number operator is represented by means of the degenerate Stirling numbers of the second kind.

This paper introduces a degenerate version of the Euler–Seidel method by incorporating a parameter $\lambda$ into the classical recurrence relation. We define a degenerate Euler–Seidel matrix associated with an initial sequence and establish corresponding $\lambda$-generalized binomial identities and generating function relations. By applying this method to the degenerate Bernoulli, Euler, and Genocchi polynomials, we derive several new combinatorial identities. This work extends the classical Euler–Seidel method to the domain of degenerate special polynomials and numbers, thus providing a new framework to study their properties.

Following Spivey's pivotal discovery of a recurrence relation for Bell numbers, significant research has emerged concerning various generalizations of Bell numbers and polynomials. For example, Kim and Kim established a Spivey-type recurrence relation specifically for degenerate Bell and Dowling polynomials. In this paper, we extend this work by deriving a probabilistic generalization of Spivey-type recurrence relations for both degenerate Bell and degenerate $r$-Bell polynomials.

In this paper, we find several expressions for the moments of the hyperbolic secant distribution and the moments of the sum of two independent distributions. They are expressed in terms of Euler and Bernoulli numbers, zeta values, an integral representation, and certain infinite series.

Assume that the moment-generating function of the random variable $Y$ exists in a neighborhood of the origin. The aim of this paper is to investigate the probabilistic type 2 Bernoulli polynomials associated with $Y$ and the probabilistic type 2 Euler polynomials associated with $Y$, along with the probabilistic type 2 cosine-Bernoulli polynomials associated with $Y$, the probabilistic type 2 sine-Bernoulli polynomials associated with $Y$, the probabilistic type 2 cosine-Euler polynomials associated with $Y$, and the probabilistic type 2 sine-Euler polynomials associated with $Y$. We deal with their properties, related identities and explicit expressions.