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We develop a unified method to study spectral determinants for several different manifolds, including spheres and hemispheres, and projective spaces. This is a direct consequence of an approach based on deriving recursion relations for the corresponding zeta functions, which we are then able to solve explicitly. Apart from new applications such as hemispheres, we also believe that the resulting formulae in the cases for which expressions for the determinant were already known are simpler and easier to compute in general, when compared to those resulting from other approaches. [ABSTRACT FROM AUTHOR]

In this work, we establish some characteristics for a sequence, A α (n , k) , including recurrence relations, generating function and inversion formula, etc. Based on the sequence, we derive, by means of the generating function approach, some transformation formulas concerning certain combinatorial numbers named after Lah, Stirling, harmonic, Cauchy and Catalan, as well as several closed finite sums. In addition, the relationship between A α (n , k) and r-Whitney–Lah numbers is established, and some formulas for the r-Whitney–Lah numbers are obtained. [ABSTRACT FROM AUTHOR]

This paper presents a number of identities for Dirichlet series and series with Stirling numbers of the first kind. As coefficients for the Dirichlet series we use Cauchy numbers of the first and second kinds, hyperharmonic numbers, derangement numbers, binomial coefficients, central binomial coefficients, and Catalan numbers.

In this work, we establish some characteristics for a sequence, Aα(n,k), including recurrence relations, generating function and inversion formula, etc. Based on the sequence, we derive, by means of the generating function approach, some transformation formulas concerning certain combinatorial numbers named after Lah, Stirling, harmonic, Cauchy and Catalan, as well as several closed finite sums. In addition, the relationship between Aα(n,k) and r-Whitney–Lah numbers is established, and some formulas for the r-Whitney–Lah numbers are obtained.

Kurokawa and Wakayama [Proc. Amer. Math. Soc. 132 (2004), pp. 935–943] defined a q-analogue of the Euler constant and proved the irrationality of certain numbers involving q-Euler constant. In this paper, we improve their results and prove the linear independence result involving q-analogue of the Euler constant. Further, we derive the closed-form of a q-analogue of the k-th Stieltjes constant \gamma _k(q). These constants are the coefficients in the Laurent series expansion of a q-analogue of the Riemann zeta function around s=1. Using a result of Nesterenko [C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), pp. 909–914], we also settle down a question of Erdős regarding the arithmetic nature of the infinite series \sum _{n\geq 1}{\sigma _1(n)}/{t^n} for any integer t > 1. Finally, we study the transcendence nature of some infinite series involving \gamma _1(2). [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Poly-Cauchy numbers with level 2 are defined by inverse sine hyperbolic functions with the inverse relation from sine hyperbolic functions. In this paper, we introduce the Stirling numbers of the first kind with level 2 in order to establish some relations with poly-Cauchy numbers with level 2. Then, we show several convolution identities of poly-Cauchy numbers with level 2. In particular, that of three poly-Cauchy numbers with level 2 can be expressed as a simple form. [ABSTRACT FROM AUTHOR]

Stirling numbers of the first kind are common in number theory and combinatorics; through Ewen's sampling formula, these numbers enter into the calculation of several population genetics statistics, such as Fu's F_s. In previous papers we have considered an asymptotic estimator for a finite sum of Stirling numbers, which enables rapid and accurate calculation of Fu's F_s. These sums can also be viewed as cumulative distribution functions, leading directly to the possibility of an inversion function, where, given a value for Fu's F_s, the goal is to solve for one of the input parameters. We solve this inversion using Newton iteration for small parameters. For large parameters, we have to extend our earlier obtained asymptotic results to solve the inversion problem asymptotically. Numerical experiments are given to show the efficiency of both solving the inversion problem and the expanded asymptotic estimator for sums of Stirling numbers. [ABSTRACT FROM AUTHOR]

Ewen’s sampling formula is a foundational theoretical result that connects probability and number theory with molecular genetics and molecular evolution; it was the analytical result required for testing the neutral theory of evolution, and has since been directly or indirectly utilized in a number of population genetics statistics. Ewen’s sampling formula, in turn, is deeply connected to Stirling numbers of the first kind. Here, we explore the cumulative distribution function of these Stirling numbers, which enables a single direct estimate of the sum, using representations in terms of the incomplete beta function. This estimator enables an improved method for calculating an asymptotic estimate for one useful statistic, Fu’s Fs. By reducing the calculation from a sum of terms involving Stirling numbers to a single estimate, we simultaneously improve accuracy and dramatically increase speed.

In this paper, we investigate a generalization of the classical Stirling numbers of the first kind by considering permutations over tuples with an extra condition on the minimal elements of the cycles. The main focus of this work is the analysis of combinatorial properties of these new objects. We give general combinatorial identities and some recurrence relations. We also show some connections with other sequences such as poly-Cauchy numbers with higher level and central factorial numbers. To obtain our results, we use pure combinatorial arguments and classical manipulations of formal power series. [ABSTRACT FROM AUTHOR]

The current paper is dedicated to proving the many properties of falling numbers and Stirling numbers. Then, these properties are implemented to construct the recurrence relations of the sums of powers σm(n) =Σ n k=1 k m [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to present various identities and computation formulas for certain classes of Apostol-type numbers and polynomials. The results of this paper contain not only the λ-Apostol-Daehee numbers and polynomials, but also Simsek numbers and polynomials, the Stirling numbers of the first kind, the Daehee numbers, and the Chu-Vandermonde identity. Furthermore, we derive an infinite series representation for the λ-Apostol-Daehee polynomials. By using functional equations containing the generating functions for the Cauchy numbers and the Riemann integrals of the generating functions for the λ-Apostol-Daehee numbers and polynomials, we also derive some identities and formulas for these numbers and polynomials. Moreover, we give implementation of a computation formula for the λ-Apostol-Daehee polynomials in Mathematica by Wolfram language. By this implementation, we also present some plots of these polynomials in order to investigate their behaviour in some randomly selected special cases of their parameters. Finally, we conclude the paper with some comments and observations on our results. [ABSTRACT FROM AUTHOR]

Ewen's sampling formula is a foundational theoretical result that connects probability and number theory with molecular genetics and molecular evolution; it was the analytical result required for testing the neutral theory of evolution, and has since been directly or indirectly utilized in a number of population genetics statistics. Ewen's sampling formula, in turn, is deeply connected to Stirling numbers of the first kind. Here, we explore the cumulative distribution function of these Stirling numbers, which enables a single direct estimate of the sum, using representations in terms of the incomplete beta function. This estimator enables an improved method for calculating an asymptotic estimate for one useful statistic, Fu's Fs. By reducing the calculation from a sum of terms involving Stirling numbers to a single estimate, we simultaneously improve accuracy and dramatically increase speed. [ABSTRACT FROM AUTHOR]

In this paper, we firstly consider extended degenerate central factorial numbers of the second kind and provide some properties of them. We then introduce unified degenerate central Bell polynomials and numbers and investigate many relations and formulas including summation formula, explicit formula and derivative property. Moreover, we derive several correlations for the fully degenerate central Bell polynomials associated with the degenerate Bernstein polynomials and the degenerate Bernoulli, Euler and Genocchi numbers. [ABSTRACT FROM AUTHOR]

In this paper, we introduce mixed coloured permutations, permutations with certain coloured cycles, and study the enumerative properties of these combinatorial objects. We derive the generating function, closed forms, recursions and combinatorial identities for the counting sequence, for mixed Stirling numbers of the first kind. In this comprehensive study, we consider further the conditions on the length of the cycles, r-mixed Stirling numbers and the connection to Bell polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we investigate p-adic q-integral (q-Volkenborn integral) on Zp of p-adic gamma function via their Mahler expansions. We also derived two q-Volkenborn integrals of p-adic gamma function in terms of q-Daehee polynomials and numbers and q-Daehee polynomials and numbers of the second kind. Moreover, we discover q-Volkenborn integral of the derivative of p-adic gamma function. We acquire the relationship between the p-adic gamma function and Stirling numbers of the first kind. We finally develop a novel and interesting representation for the p-adic Euler constant by means of the q-Daehee polynomials and numbers. [ABSTRACT FROM AUTHOR]

We set some correlations between Boole polynomials and p-adic gamma function in conjunction with p-adic Euler contant. We develop diverse formulas for p-adic gamma function by means of their Mahler expansion and fermionic p-adic integral on Zp. Also, we acquire two fermionic p-adic integrals of p-adic gamma function in terms of Boole numbers and polynomials. We then provide fermionic p-adic integral of the derivative of p-adic gamma function and a representation for the p-adic Euler constant by means of the Boole polynomials. Furthermore, we investigate an explicit representation for the aforesaid constant covering Stirling numbers of the first kind. [ABSTRACT FROM AUTHOR]

In this paper, we primarily consider a generalization of the fermionic p -adic q -integral on ℤ p including the parameters α and β and investigate its some basic properties. By means of the foregoing integral, we introduce two generalizations of q -Changhee polynomials and numbers as q -Changhee polynomials and numbers with weight (α , β) and q -Changhee polynomials and numbers of second kind with weight (α , β). For the mentioned polynomials, we obtain new and interesting relationships and identities including symmetric relation, recurrence relations and correlations associated with the weighted q -Euler polynomials, λ -Stirling numbers of the second kind and Stirling numbers of first and second kinds. Then, we discover multifarious relationships among the two types of weighted q -Changhee polynomials and p -adic gamma function. Also, we compute the weighted fermionic p -adic q -integral of the derivative of p -adic gamma function. Moreover, we give a novel representation for the p -adic Euler constant by means of the weighted q -Changhee polynomials and numbers. We finally provide a quirky explicit formula for p -adic Euler constant. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new formula for hyper-Fibonacci numbers Fn[k] wherein the coefficients (related to Stirling numbers of the first kind) of the polynomial ingredient pk(n) are determined. As an application we investigate the number of occurrences of positive integers among Fn[k] and determine all the solutions in nonnegative integers x and y to the Diophantine equation Fx[k]= Fy[ℓ], where 0 ≤ k < ℓ ≤ 70. Moreover, we prove that if l is fixed, then Fx[k] = Fy[ℓ] has finitely many effectively computable solutions in the nonnegative integers x, y, and k ≤ ℓ. [ABSTRACT FROM AUTHOR]

In this paper we analyze a recently proposed impartial combinatorial ruleset that is played on a permutation of the set n. We call this ruleset STIRLING SHAVE. A procedure utilizing the ordinal sum operation is given to determine the nim value of a given normal play position. Additionally, we enumerate the number of permutations of n which are P-positions. The formula given involves the Stirling numbers of the first-kind. We also give a complete analysis of the Misère version of STIRLING SHAVE using Conway’s genus theory. An interesting by-product of this analysis is insight into how the ordinal sum operation behaves in Misère Play. [ABSTRACT FROM AUTHOR]

In the paper, the authors present an explicit form for a family of inhomoge- neous linear ordinary differential equations, find a more significant expression for all derivatives of a function related to the solution to the family of inhomo- geneous linear ordinary differential equations in terms of the Lerch transcen- dent, establish an explicit formula for computing all derivatives of the solution to the family of inhomogeneous linear ordinary differential equations, acquire the absolute monotonicity, complete monotonicity, the Bernstein function property of several functions related to the solution to the family of inhomo- geneous linear ordinary differential equations, discover a diagonal recurrence relation of the Stirling numbers of the first kind, and derive an inequality for bounding the logarithmic function. [ABSTRACT FROM AUTHOR]

The main motivation of this paper is to investigate some derivative properties of the generating functions for the numbers Y n ( λ ) and the polynomials Y n ( x ; λ ) , which were recently introduced by Simsek [30] . We give functional equations and differential equations (PDEs) of these generating functions. By using these functional and differential equations, we derive not only recurrence relations, but also several other identities and relations for these numbers and polynomials. Our identities include the Apostol–Bernoulli numbers, the Apostol–Euler numbers, the Stirling numbers of the first kind, the Cauchy numbers and the Hurwitz–Lerch zeta functions. Moreover, we give hypergeometric function representation for an integral involving these numbers and polynomials. Finally, we give infinite series representations of the numbers Y n ( λ ) , the Changhee numbers, the Daehee numbers, the Lucas numbers and the Humbert polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a general symmetric identity involving the degenerate Euler-tangent mixed-type polynomials and sums of generalized falling factorials. We use this identity to describe some combinatorial relations between these polynomials and generalized factorial alternating sums. Finally, we observe an interesting phenomenon of "scattering" of the zeros of degenerate Euler-tangent mixed-type polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we investigate two kinds of Euler sums that involve the generalized harmonic numbers with arbitrary depth. These sums establish numerous summation formulas including the special values of Arakawa-Kaneno zeta functions and a new formula of multiple zeta values of height one as examples. [ABSTRACT FROM AUTHOR]

We provide completely effective error estimates for Stirling numbers of the first and second kinds, denoted by s( n, m) and S( n, m), respectively. These bounds are useful for values of $${m\geq n-O(\sqrt{n})}$$ . An application of our Theorem 3.2 yields, for example, The bounds are obtained via Chen-Stein Poisson approximation, using an interpretation of Stirling numbers as the number of ways of placing non-attacking rooks on a chess board. As a corollary to Theorem 3.2, summarized in Proposition 2.4, we obtain two simple and explicit asymptotic formulas, one for each of s( n, m) and S( n, m), for the parametrization $${m = n-t {n^a}, 0 \leq a \leq \frac{1}{2}}$$ . These asymptotic formulas agree with the ones originally observed by Moser and Wyman in the range $${0 < a < \frac{1}{2}}$$ , and they connect with a recent asymptotic expansion by Louchard for $${\frac{1}{2} < a < 1}$$ , hence filling the gap at $${a = \frac{1}{2}}$$ . We also provide a generalization applicable to rook and file numbers. [ABSTRACT FROM AUTHOR]

In this paper we solve three open problems and a conjecture related to the calculations of some classes of multiple series posed by Furdui in [1].

We find an explicit formula for computing the Bernoulli numbers of the second kind in terms of the signed Stirling numbers of the first kind. [ABSTRACT FROM AUTHOR]

General relations expressing factorial moments in terms of raw moments, raw moments in terms of factorial moments, factorial moments in terms of central moments, and central moments in terms of factorial moments are derived. Examples are given. [ABSTRACT FROM AUTHOR]

Stirling numbers of the first kind has some interesting interpretations. In this short paper, we exhibit an elementary deduction of an identity for Snmobtained by Petersen-Varchenko. [ABSTRACT FROM AUTHOR]