Note on $ r $-central Lah numbers and $ r $-central Lah-Bell numbers

<abstract><p>The $ r $-Lah numbers generalize the Lah numbers to the $ r $-Stirling numbers in the same sense. The Stirling numbers and the central factorial numbers are one of the important tools in enumerative combinatorics. The $ r $-Lah number counts the number of partitions of a set with $ n+r $ elements into $ k+r $ ordered blocks such that $ r $ distinguished elements have to be in distinct ordered blocks. In this paper, the $ r $-central Lah numbers and the $ r $-central Lah-Bell numbers ($ r\in \mathbb{N} $) are introduced parallel to the $ r $-extended central factorial numbers of the second kind and $ r $-extended central Bell polynomials. In addition, some identities related to these numbers including the generating functions, explicit formulas, binomial convolutions are derived. Moreover, the $ r $-central Lah numbers and the $ r $-central Lah-Bell numbers are shown to be represented by Riemann integral, respectively.</p></abstract>

Expectation Identity of the Hypergeometric Distribution and Its Application in the Calculations of High-Order Origin Moments

We provide a novel method to analytically calculate the high-order origin moments of a hypergeometric distribution, that is, the expectation identity method. First, the expectation identity of the hypergeometric distribution is discovered and summarized in a theorem. After that, we analytically calculate the first four origin moments of the hypergeometric distribution by using the expectation identity. Furthermore, we analytically calculate the general kth (k=1,2,…) origin moment of the hypergeometric distribution by using the expectation identity, and the results are summarized in a theorem. Moreover, we use the general kth origin moment to validate the first four origin moments of the hypergeometric distribution. Next, the coefficients of the first ten origin moments of the hypergeometric distribution are summarized in a table containing Stirling numbers of the second kind. Moreover, the general kth origin moment of the hypergeometric distribution by using the expectation identity is restated by another theorem involving Stirling numbers of the second kind. Finally, we provide some numerical and theoretical results.

Construction of Type 2 Poly-Changhee Polynomials and Its Applications

In this paper, we introduce type 2 poly-Changhee polynomials by using the polyexponential function. We derive some explicit expressions and identities for these polynomials, and we also prove some relationships between poly-Changhee polynomials and Stirling numbers of the first and second kind. Also, we introduce the unipoly-Changhee polynomials by employing unipoly function and give multifarious properties. Furthermore, we provide a correlation between the unipoly-Changhee polynomials and the classical Changhee polynomials.

Polynomials, numerical triangles and sequences associated with the probability distribution of the hyperbolic cosine type for odd values of the parameter

The article introduces a new class of polynomials that first appeared in the probability distribution density function of the hyperbolic cosine type. With an integer change in one of the parameters of this distribution, polynomials in the form of a product of positive factors are written out with an increasing degree. Earlier, the author found a connection between the distribution of the hyperbolic cosine type and numerical sets, in particular, in the simplest cases with the triangle of coefficients of Bessel polynomials, the triangle of Stirling numbers, sequences of coefficients in the expansion of various functions, etc. Also from the distribution formed numerous numerical sequences, both new and widely known. Consideration of polynomials separately from the density function made it possible to reconstruct numerical sets of coefficients, ordered in the form of numerical triangles and numerical sequences. The connections between the elements of the sets are established. Among the sequences obtained, in the simplest cases, there are those known from others, for example, physical problems. However, the overwhelming majority of the found number sets have not been encountered earlier in the literature. The obvious applications of this research are number theory and algebra. And the interdisciplinarity of the results indicates the possibility of applications and enhances their practical significance in other areas of knowledge.

Polynomials and number sets associated with the probability distribution of the hyperbolic cosine type for even values of the parameter

The polynomials used in the formation of the probability distribution density function of the hyperbolic cosine type are investigated. Earlier, on the basis of a hyperbolic cosine distribution, the author obtained numerical sets, among which not only new ones, but also, for example, the triangle of Stirling numbers, the triangle of the coefficients of Bessel polynomials, sequences of coefficients in the expansion of various functions, etc. In this paper, depending on the natural parameter m and the real distribution parameter β , a new class of polynomials is obtained. For even and odd m , the polynomials are constructed using similar, but different formulas. The article presents polynomials for even values m . Structurally, polynomials consist of quadratic factors. The coefficients of the polynomials, ordered by m , form numerical triangles depending on β . Some relations are found between the coefficients. From the numerical triangles, a set of numerical sequences is obtained, which for integers β are integers. Also, polynomials with respect to x turn out to be polynomials with respect to β . With this interpretation the variable acts as a parameter. New numerical triangles and sequences for different x were found. The overwhelming majority of the obtained numerical sequences are new. The class of polynomials arising from problems of probability theory indicates the possibility of applying the results.

New Properties on Degenerate Bell Polynomials

The aim of this paper is to study the degenerate Bell numbers and polynomials which are degenerate versions of the Bell numbers and polynomials. We derive some new identities and properties of those numbers and polynomials that are associated with the degenerate Stirling numbers of both kinds.

Maclaurin's series expansions of real powers of inverse (hyperbolic) cosine and sine functions with applications

In the paper, by means of the Faa di Bruno formula, with the help of explicit formulas for special values of the Bell polynomials of the second kind with respect to a specific sequence, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes Maclaurin's series expansions for real powers of the inverse cosine (sine) function and the inverse hyperbolic cosine (sine) function. By applying different series expansions for the square of the inverse cosine function, the author not only finds infinite series representations of the circular constant Pi and its square, but also derives two combinatorial identities involving central binomial coefficients.