Abstract Let Y be a random variable such that the moment generating function of exists in neighborhood origin. The aim this paper is to study probabilistic versions degenerate Fubini polynomials and order r , namely probabilisitc associated with . We derive some properties, explicit expressions, certain identities recurrence relations for those polynomials. As special cases we treat gamma parameters $$\alpha ,\beta &gt; 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow>...

In recent years, both degenerate versions and probabilistic extensions of many special numbers polynomials have been explored. For instance, Bernstein were investigated earlier. Assume that Y is a random variable whose moment generating function exists in neighbourhood the origin. The aim this paper to study associated with which are extension version Y. We derive several explicit expressions certain related identities for those polynomials. addition, we treat cases Poisson variable,...

The aim of this paper is to research the structural properties Fibonacci polynomials and numbers obtain some identities. To achieve purpose, we first introduce a new second-order nonlinear recursive sequence. Then, our main results by using sequence, power series, combinatorial methods.

Abstract Recently, the n th Lah–Bell number was defined as of ways a set elements can be partitioned into nonempty linearly ordered subsets for any nonnegative integer . Further, natural extensions numbers, polynomials are defined. We study with and without help umbral calculus. Notably, we use three different formulas in order to express various known families such higher-order Bernoulli poly-Bernoulli terms polynomials. In addition, obtain several properties

Let Y be a random variable such that the moment generating function of exists in neighborhood origin. The aim this paper is to study probabilistic versions degenerate Fubini polynomials and order $r$, namely probabilisitc associated with r Y. We derive some properties, explicit expressions, certain identities recurrence relations for those polynomials.

Recently, degenerate harmonic numbers and hyperharmonic are introduced by Kim-Kim. In this paper, we study the series involving Stirling investigate those properties.

Assume that is Y a random variable whose moment generating function exists in neighbourhood of the origin. The aim this paper to study probabilistic poly-Bernoulli numbers associated with Y, as extensions numbers. We derive explicit expressions, some related identities and symmetric relation for those also investigate expressions modified probabilisitc Bernoulli which are slightly different from probabilisitic Y. As special cases we treat Poisson, gamma variables.

We use the elementary and analytic methods properties of Chebyshev polynomials to study computational problem reciprocal sums one‐kind give several interesting identities for them. At same time, we also a general method this kind sums.

Abstract In this paper, we introduce one discrete random variable, namely the negative λ -binomial variable. We deduce expectation of also get variance and explicit expression for moments

Abstract The classical Dedekind sums appear in the transformation behavior of logarithm eta-function under substitutions from modular group. and their generalizations are defined terms Bernoulli functions generalizations, shown to satisfy some reciprocity relations. In contrast, Dedekind-type DC (Daehee Changhee) Euler generalizations. purpose this paper is introduce poly-Dedekind-type sums, which obtained by replacing function poly-Euler arbitrary indices, show that those satisfy, among...

Recently, Kim-Kim introduced the truncated degenerate Bell polynomials and numbers. In this paper, we introduce Lah-Bell We obtain some identities, recurrence relations properties. Furthermore, also modified numbers, numbers derive properties identities involving with those

Abstract Apostol considered generalized Dedekind sums by replacing the first Bernoulli function appearing in any functions and derived a reciprocity relation for them. Recently, poly-Dedekind were introduced type 2 poly-Bernoulli of arbitrary indices shown to satisfy relation. In this paper, we consider other that are obtained indices. We derive these sums.

Catalan-Daehee numbers and polynomials, generating functions of which can be expressed as p-adic Volkenborn integrals on Zp, were studied previously. The aim this paper is to introduce qanalogues the polynomials with help q-integrals Zp. We derive, among other things, some explicit expressions for q-analogues polynomials.

Recently, Kim-Kim introduced the lambda-umbral calculus, in which lambda-Sheffer sequences occupy central position. In this paper, we introduce fully degenerate Bell and Dowling polynomials, investigate some properties identities relating to those polynomials with help oflambda-umbral calculus. Here note that poynomials are respectively versions of latters natural extension Whitney numbers second kind.

Protein structure comparison is one of the most important problems in computational biology and plays a key role protein prediction. A new method based on shape distribution Calpha representation backbone proposed for evaluating structural similarity between pairs. The completely independent sequence information. By examining 3D proteins data from Data Bank (PDB) using method, strong detected pairs with close structures functions. It provides helpful to measure pairspsila when not enough.

Abstract Dedekind sums and their generalizations are defined in terms of Bernoulli functions generalizations. As a new generalization the sums, degenerate poly-Dedekind which obtained from by replacing poly-Bernoulli arbitrary indices introduced this article shown to satisfy reciprocity relation.

The application of big data accurate portrait marketing is a kind technology and analysis methods, through the collection, collation, user behavior preferences other data, segmentation portrait, in order to achieve more strategy res

Based on the degenerate unipoly-Euler functions and type 2 functions, we first introduce two types of unipoly-Dedekind DC (Daehee-Changhee) sums. By using definitions unipoly Stirling numbers kind second kind, Genocchi polynomials, numbers, obtain several combinatorial identities properties functions. The sums are shown to satisfy reciprocity relations.

Let $q>2$ be an integer, $n\geqslant2$ a fixed integer with $(n,q)=1$ , ψ non-principal Dirichlet character modq. An upper bound estimate for sums of the form $$\sum_{a\in\textit{C}(1,q)}\psi(a) $$ is given, where $\mathcal{C}(1,q)=\{a \mid 1\leqslant a\leqslant q-1, a\overline{a}\equiv1 (\bmod q), n\nmid(a+\overline{a})\}$ .

Dedekind-type DC sums and their properties are defined in terms of Euler functions. Ma et al. recently introduced poly-Dedekind-type demonstrated that they satisfy a reciprocity relation. In this paper, we introduce the degenerate poly-Euler polynomials numbers, also consider relations sums. Equivalently, several identities functions obtained.