Recently, the Bell-based Stirling polynomials of the second kind and the Bell-based Bernoulli polynomials [U. Duran, S. Araci, M. Acikgoz, Axioms, 10 (2021), 23 pages] have been considered, and some of their properties and applications in umbral calculus have been derived and analyzed. In this work, a degenerate form of the Bell-based Stirling polynomials of the second kind is defined, and several fundamental properties and formulas for these polynomials are investigated and presented in detail. Then, a degenerate form of the Bell-based Bernoulli polynomials of order a is defined and a plenty of their properties are examined in different aspects. Several correlations with other polynomials and numbers in literature, symmetric identities, implicit summation formulas, derivative properties and addition formulas for the mentioned new polynomials are derived in detail, and some special cases of these results are investigated. Also, the degenerate Bell-based Bernoulli polynomials of order e are studied in A-umbral calculus and interesting relations and formulas are developed. Furthermore, the application of A-umbral calculus to Bell-based degenerate Bernoulli polynomials of order e shows a correlation with higher-order degenerate derangement polynomials. Finally, a representation of the degenerate differential operator on the degenerate Bell-based Bernoulli polynomials of order e is provided.

Our aim of this paper was to introduce a new construction of probabilistic Hermite polynomials based on moment generating functions. By using this generating function, we derived several new relations and formulas among the aforementioned polynomials and other types of probabilistic special number sequences and polynomials, such as the probabilistic Stirling numbers of the second kind, probabilistic Bernoulli polynomials of higher order, probabilistic Bernstein polynomials, and probabilistic Euler polynomials of higher order. By selecting special random variables, including Poisson, Uniform, Gamma, Geometric, Exponential, and Normal random variables, we showed that the generating function of probabilistic Hermite polynomials yields distinct and unique generating functions, which lead to new relations among other types of special numbers and polynomials, as presented in the application section of this paper.

Fibonacci extensions of several special polynomials, including Fibonacci-Bernoulli, Fibonacci-harmonic, Fibonacci-Euler, and Fibonacci-Hermite polynomials, have recently been studied, and numerous properties and relations of these polynomials have been thoroughly examined utilizing the content of the golden calculus. This paper aims to consider the generating functions of the Fibonacci-Gould-Hopper polynomials and the Gould-Hopper-based Fibonacci-Frobenius-sigmoid polynomials, from which we derive several beneficial relations and properties. These include explicit formulas, summation formulas, correlation formulas with the new and old Fibonacci-type polynomials, symmetric properties, recurrence relation, addition formulas, golden derivative properties, and golden integral representation for these polynomials. Moreover, graphical illustrations of the Fibonacci-Gould-Hopper polynomials and the Gould-Hopper-based Fibonacci-Frobenius-sigmoid polynomials are presented. Their numerical analyses are used to validate theoretical results and reveal distinctive scattering patterns in the distribution of their zeros across the complex plane, offering insights into their underlying analytic structure. Furthermore, interesting patterns in the zeros (real and complex zeros) distributions of these two new families of polynomials are examined and drawn, forming 2D and 3D structures. In addition, the approximate real and complex zeros of the mentioned polynomials for some special cases are presented in four tables. Lastly, four conjectures about the zeros of these polynomials are given.

In this paper, we introduce a new class of polynomials, called probabilistic q-Bernstein polynomials, alongside their generating function. Assuming Y is a random variable satisfying moment conditions, we use the generating function of these polynomials to establish new relations. These include connections to probabilistic Stirling numbers of the second kind and higher-order probabilistic Bernoulli polynomials associated with Y. Additionally, we derive recurrence and differentiation properties for probabilistic q-Bernstein polynomials. Utilizing Leibniz's formula, we give an identity for the generating function of these polynomials. In the latter part of the paper, we explore applications by choosing appropriate random variables such as Poisson, Bernoulli, Binomial, Geometric, Negative Binomial, and Uniform distributions. This allows us to derive relationships among probabilistic q-Bernstein polynomials, Bell polynomials, Stirling numbers of the second kind, higher-order Frobenius-Euler numbers, and higher-order Bernoulli polynomials. We also present p-adic q-integral and fermionic p-adic q-integral representations for probabilistic q-Bernstein polynomials.

We first review and analyze the golden integral and its definitions and some properties. Then we introduce a new generalization of the Hermite polynomials via the golden exponential function (called Fibonacci-Hermite polynomials) and investigate several properties and relations. We derive some explicit and implicit summation formulas for mentioned polynomials. Then, we analyze derivative properties and provide a higher-order difference equation of the Fibonacci-Hermite polynomials. Moreover, we examine a recurrence relation and integral representation. In addition, we obtain some properties of Fibonacci-Bernstein polynomials. Lastly, we obtain a correlation between the Fibonacci-Hermite polynomials and the Fibonacci-Bernstein polynomials

In this paper, we introduce a new family of Stirling polynomials of the second kind, Bell polynomials, bivariate Bell polynomials, Bernoulli polynomials of higher order, and Euler polynomials of higher order arising from the Kaniadakis calculus viewpoint. We refer to each of them as κ-polynomials. Through the defined concepts of Kaniadakis calculus, we derive explicit formulas, summation formulas, and addition formulas for the polynomials discussed in the present paper. We also present the Volkenborn integral and the fermionic p-adic integral representations in terms of the κ-Stirling polynomials of the second kind, bivariate κ-Bell polynomials, κ-Bernoulli polynomials of higher order, and κ-Euler polynomials of higher order. We establish some formulae, including old and new polynomials. Finally, we investigate determinantal representations for the κ-Euler polynomials and the κ-Bernoulli polynomials.

In this paper, we introduce the Fourier series expansion of the generating function for Bernstein polynomials. We also present series formulas for the generating function of Bernoulli polynomials. Furthermore, we establish novel formulae between these series and Euler polynomials as well as Zeta-type functions. The exploration of these connections sheds light on the intricate relationships among these fundamental mathematical constructs. Through these discoveries, we deepen our understanding of the interplay between various polynomial families and associated mathematical functions. These findings contribute to the broader landscape of mathematical analysis and offer insights into the rich structure underlying theory of special functions.

In this paper, we introduce the modified q-Genocchi polynomials, investigate their properties, and give their generating function. We obtain a number of new relations and properties for q-Genocchi polynomials, such as addition formula, explicit formula, derivative formula, and multiplication formula. As an application, a new q-analogue of Genocchi zeta function is presented by applying the Mellin transform to the generating function of the modified q-Genocchi polynomials. Finally, we define the q-Genocchi zeta-type functions and then prove their interpolation by the modified q-Genocchi polynomials at negative integers.

In this paper, we introduce the probabilistic Bernstein polynomials and derive new and interesting correlations among several special functions and special number sequences such as Euler polynomials, Bernoulli polynomials of higher order, Frobenius-Euler polynomials of higher order, Stirling numbers of the second kind and Bell polynomials subject to several special random variables.

In this paper, the Gould-Hopper based fully degenerate type2 poly-Stirling polynomials of the first kind with a q parameter are considered and some of their diverse identities and properties are investigated.Then, the Gould-Hopper based fully degenerate type2 poly-Bernoulli polynomials with a q parameter are introduced and some of their properties are analyzed and derived.Furthermore, several formulas and relations covering implicit summation formulas, recurrence relations and symmetric property are attained.

The study of expansions of certain mock theta functions in special functions theory has a long and quite significant history. Motivated by recent correlations between $ q $-series and mock theta functions, we establish a new $ q $-series transformation formula and derive the double-sum expansions for mock theta functions. As an application, we state new double-sum representations for certain mock theta functions.

In this paper, we consider unified Gould-Hopper based Apostol-type polynomials and investigate some of their formulas including several implicit summation formulae and some symmetric identities by the series manipulation method. Moreover, we acquire several new results for unified Gould-Hopper based Apostol-type polynomials using appropriate operational rules.

In this paper, we consider Bell-based Stirling polynomials of the second kind and derive some useful relations and properties including some summation formulas related to the Bell polynomials and Stirling numbers of the second kind. Then, we introduce Bell-based Bernoulli polynomials of order α and investigate multifarious correlations and formulas including some summation formulas and derivative properties. Also, we acquire diverse implicit summation formulas and symmetric identities for Bell-based Bernoulli polynomials of order α. Moreover, we attain several interesting formulas of Bell-based Bernoulli polynomials of order α arising from umbral calculus.

In this study, we introduce Bell-based Genocchi polynomials of order α and then derive multifarious correlations and formulas including some implicit summation formulas and derivative properties.

In this paper, we derive some new symmetric properties of k-Fibonacci numbers by making use of symmetrizing operator. We also give some new generating functions for the products of some special numbers such as k-Fibonacci numbers, k-Pell numbers, Jacobsthal numbers, Fibonacci polynomials and Chebyshev polynomials.

In this paper, we introduce a new operator in order to derive some new symmetric properties of k-Fibonacci and k-Lucas numbers and Fibonacci polynomials. By making use of the new operator defined in this paper, we give some new generating functions for k-Fibonacci and Pell numbers and Fibonacci polynomials.

In this paper, we give some new generating functions of the products of (p, q)-Fibonacci numbers, (p, q) -Lucas numbers, (p, q)-Pell numbers, $$\left( p,q\right)$$ -Pell Lucas numbers, (p, q)-Jacobsthal numbers, and (p, q)-Jacobsthal Lucas numbers with 2-orthogonal Chebyshev polynomials and trivariate Fibonacci polynomials.

In this study, we consider the truncated degenerate Frobenius-Euler polynomials.Then we examine diverse properties and formulas covering addition formulas, correlations and derivation property.Then, we derive some interesting implicit summation formulas.