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The relationship between a stable multivariable polynomial p(z) and the Fourier coefficients of its spectral density function 1 / | p (z) | 2 , is further investigated. In this paper we focus on the radial asymptotics of the Fourier coefficients for a specific choice of a two variable polynomial. Hypergeometric functions appear in the analysis, and new results are derived for these as well. [ABSTRACT FROM AUTHOR]

We study several special functions in Ramanujan's generalized modular equation with signature 3. Some sharp inequalities for these functions, including the estimates for the solution of Ramanujan's generalized modular equation with signature 3 and triplication inequality for the generalized Grötzsch ring function with two parameters, are derived. [ABSTRACT FROM AUTHOR]

Given complex parameters x, ν, α, β and γ ∉ − N , consider the infinite lower triangular matrix A (x , ν ; α , β , γ) with elements A n , k (x , ν ; α , β , γ) = (− 1) k n + α k + α ⋅ F (k − n , − (β + n) ν ; − (γ + n) ; x) for 1 ⩽ k ⩽ n , depending on the Hypergeometric polynomials F (− n , ⋅ ; ⋅ ; x) , n ∈ N ∗ . After stating a general criterion for the inversion of infinite matrices in terms of associated generating functions, we prove that the inverse matrix B (x , ν ; α , β , γ) = A (x , ν ; α , β , γ) − 1 is given by B n , k (x , ν ; α , β , γ) = (− 1) k n + α k + α ⋅ γ + k β + k F (k − n , (β + k) ν ; γ + k ; x) + β − γ β + k F (k − n , (β + k) ν ; 1 + γ + k ; x) for 1 ⩽ k ⩽ n , thus providing a new class of linear inversion formulas. Functional relations for the generating functions of related sequences S and T, that is, T = A (x , ν ; α , β , γ) S ⟺ S = B (x , ν ; α , β , γ) T , are also provided. [ABSTRACT FROM AUTHOR]

In the present article, we introduce and study a new family F Σ (δ , λ , m , n , r) of normalized analytic and bi-univalent functions associating λ -pseudo functions with Sakaguchi type functions by using the Horadam polynomials. We obtain upper bounds for the initial Taylor–Maclaurin coefficients | a 2 | and | a 3 | . Further we obtain the Fekete–Szegö inequality for functions in the family F Σ (δ , λ , m , n , r) which we have introduced here. We also indicate several certain special cases and consequences for our results. [ABSTRACT FROM AUTHOR]

We showcase a collection of practical strategies to deal with a problem arising from an analysis of integral estimators derived via quasi-Monte Carlo methods. The problem reduces to a triple binomial sum, thereby enabling us to open up the holonomic toolkit, which contains tools such as creative telescoping that can be used to deduce a recurrence satisfied by the sum. While applying these techniques, a host of issues arose that partly needed to be resolved by hand. In other words, no creative telescoping implementation currently exists that can resolve all these issues automatically. Thus, we felt the need to compile the different strategies we tried and the difficulties that we encountered along the way. In particular, we highlight the necessity of the certificate in these computations and how its complexity can greatly influence the computation time. [ABSTRACT FROM AUTHOR]

Our aim is to study and investigate the family of (p , q) -extended (incomplete and complete) elliptic-type integrals for which the usual properties and representations of various known results of the (classical) elliptic integrals are extended in a simple manner. This family of elliptic-type integrals involves a number of special cases and has a connection with (p , q) -extended Gauss' hypergeometric function and (p , q) -extended Appell's double hypergeometric function F 1 . Turán-type inequalities including log-convexity properties are proved for these (p , q) -extended complete elliptic-type integrals. Further, we establish various Mellin transform formulas and obtain certain infinite series representations containing Laguerre polynomials. We also obtain some relationship between these (p , q) -extended elliptic-type integrals and Meijer G-function of two variables. Moreover, we obtain several connections with (p , q) -extended beta function as special values and deduce numerous differential and integral formulas. In conclusion, we introduce (p , q) -extension of the Epstein–Hubbell (E-H) elliptic-type integral. [ABSTRACT FROM AUTHOR]

In this study, a new family of discrete multiple orthogonal polynomials, namely, ω-multiple Meixner polynomials of the second kind, where ω is a positive real number which is introduced. Some structural properties for these polynomials such as raising operator, Rodrigue's type formula and explicit representation are obtained. Generating function for ω-multiple Meixner polynomials of the second kind and several consequences using this generating function for these polynomials are derived. A lowering operator for ω-multiple Meixner polynomials of the second kind which will be helpful for obtaining difference equation is derived. By combining the lowering operator and the raising operator the difference equation having the ω-multiple Meixner polynomials of the second kind as a solution is obtained. A third-order explicit difference equation for ω-multiple Meixner polynomials of the second kind is given as a corollary. It is proven that when ω = 1 , the obtained results coincide with the existing results for multiple Meixner polynomials of the second kind. In the last section, the case when ω = 5 / 3 is studied and for the 5/3-multiple Meixner polynomials of the second kind the explicit form, generating function and the third-order difference equation is given. [ABSTRACT FROM AUTHOR]

In this article, we find finite field analogues of certain identities satisfied by the classical 4 F 3 -hypergeometric series and Appell series. As an application, we find new summation and product formulas satisfied by the Gaussian hypergeometric series. For example, we express a 4 F 3 -Gaussian hypergeometric series as a sum of two 2 F 1 -Gaussian hypergeometric series. We also find two identities expressing 4 F 3 -Gaussian hypergeometric series as a product of two 2 F 1 -Gaussian hypergeometric series. As another application, we find a special value of a 4 F 3 -Gaussian hypergeometric series using the summation formula. [ABSTRACT FROM AUTHOR]

The simple (linear) birth-and-death process is a widely used stochastic model for describing the dynamics of a population. When the process is observed discretely over time, despite the large amount of literature on the subject, little is known about formal estimator properties. Here we will show that its application to observed data is further complicated by the fact that numerical evaluation of the well-known transition probability is an ill-conditioned problem. To overcome this difficulty we will rewrite the transition probability in terms of a Gaussian hypergeometric function and subsequently obtain a three-term recurrence relation for its accurate evaluation. We will also study the properties of the hypergeometric function as a solution to the three-term recurrence relation. We will then provide formulas for the gradient and Hessian of the log-likelihood function and conclude the article by applying our methods for numerically computing maximum likelihood estimates in both simulated and real dataset. [ABSTRACT FROM AUTHOR]

We give a new method to prove in a uniform and easy way various transformation formulas for Gauss hypergeometric functions. The key is Jacobi's canonical form of the hypergeometric differential equation. Analogy for q-hypergeometric functions is also studied. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to introduce some special families of holomorphic and Al-Oboudi type bi-univalent functions related to k-Fibonacci numbers involving modified Sigmoid activation function ϕ (s) = 2 1 + e - s , s ≥ 0 in the open unit disc D . We investigate the upper bounds on initial coefficients for functions of the form g ϕ (z) = z + ∑ j = 2 ∞ ϕ (s) d j z j , in these newly introduced special families and also discuss the Fekete–Szegö problem. Some interesting consequences of the results established here are also indicated. [ABSTRACT FROM AUTHOR]

The objective of this article is to evaluate unified fractional integrals and derivative formulas involving the incomplete τ -hypergeometric function. These integrals and derivatives are further applied in proving theorems on Marichev–Saigo–Maeda operators of fractional integration and differentiation. The results are expressed in terms of the generalized Gauss hypergeometric functions (Fox–Wright function). Corresponding assertions in terms of Saigo, Erdélyi–Kober, Riemann–Liouville, and Weyl type of fractional integrals and derivatives are presented. Also, we develop their composition formula by applying the Beta and Laplace transforms. Further, we point out also their relevance. [ABSTRACT FROM AUTHOR]

The Voros coefficient of the Gauss hypergeometric differential equation with a large parameter is defined for the origin and its explicit form and the details of derivation are given. [ABSTRACT FROM AUTHOR]

We investigate interlacing properties of zeros of Laguerre polynomials L n (α) (x) and L n + 1 (α + k) (x) , α > − 1 , where n ∈ N and k ∈ { 1 , 2 }. We prove that, in general, the zeros of these polynomials interlace partially and not fully. The sharp t-interval within which the zeros of two equal degree Laguerre polynomials L n (α) (x) and L n (α + t) (x) are interlacing for every n ∈ N and each α > − 1 is 0 < t ≤ 2 , [Driver K, Muldoon ME. Sharp interval for interlacing of zeros of equal degree Laguerre polynomials. J Approx Theory, to appear.], and the sharp t-interval within which the zeros of two consecutive degree Laguerre polynomials L n (α) (x) and L n − 1 (α + t) (x) are interlacing for every n ∈ N and each α > − 1 is 0 ≤ t ≤ 2 , [Driver K, Muldoon ME. Common and interlacing zeros of families of Laguerre polynomials. J Approx Theory. 2015;193:89–98]. We derive conditions on n ∈ N and α, α > − 1 that determine the partial or full interlacing of the zeros of L n (α) (x) and the zeros of L n (α + 2 + k) (x) , k ∈ { 1 , 2 }. We also prove that partial interlacing holds between the zeros of L n (α) (x) and L n − 1 (α + 2 + k) (x) when k ∈ { 1 , 2 } , n ∈ N and α > − 1. Numerical illustrations of interlacing and its breakdown are provided. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a (p , ν) {(p,\nu)} -extension of Srivastava's triple hypergeometric function H B ⁢ (⋅) {H_{B}(\,\cdot\,)} , by using the extended beta function B p , ν ⁢ (x , y) {B_{p,\nu}(x,y)} introduced in [R. K. Parmar, P. Chopra and R. B. Paris, On an extension of extended beta and hypergeometric functions, J. Class. Anal. 11 2017, 2, 91–106]. We give some of the main properties of this extended function, which include several integral representations involving Exton's hypergeometric function, the Mellin transform, a differential formula, recursion formulas and a bounded inequality. [ABSTRACT FROM AUTHOR]

The aim of this note is to provide some new Gaussian hypergeometric summation formulae. These are further used to obtain certain new expressions for the product of hypergeometric series. Already obtained results by Bailey, Choi and Rathie and Qureshi et al. follow special cases of our main findings. [ABSTRACT FROM AUTHOR]

A new family of p-Bernoulli numbers and polynomials was introduced by Rahmani (J. Number Theory 157:350–366, 2015) with the help of the Gauss hypergeometric function. Motivated by that paper and in the light of the recent interests in finding degenerate versions, we construct the generalized degenerate Bernoulli numbers and polynomials by means of the Gauss hypergeometric function. In addition, we construct the degenerate type Eulerian numbers as a degenerate version of Eulerian numbers. For the generalized degenerate Bernoulli numbers, we express them in terms of the degenerate Stirling numbers of the second kind, of the degenerate type Eulerian numbers, of the degenerate p-Stirling numbers of the second kind and of an integral on the unit interval. As to the generalized degenerate Bernoulli polynomials, we represent them in terms of the degenerate Stirling polynomials of the second kind. [ABSTRACT FROM AUTHOR]

We prove that, for p ∈ (1 , ∞) and β ∈ R , the function x ↦ β − log ⁡ 1 − x p K p (x p) is strictly concave on (0 , 1) if and only if β ≥ λ (p) := 2 p (p 2 − 2 p + 2) (p − 1) (2 p 2 − 3 p + 3) , where K p represents the generalized complete p-elliptic integrals of the first kind defined by K p (r) := ∫ 0 π p / 2 d θ (1 − r p sin p p ⁡ θ) 1 − 1 / p , where π p := 2 p B (1 / p , 1 − 1 / p) , π 2 = π , and sin p is the generalized sine function, with sin 2 = sin. This extends the recently obtained corresponding result for the case that p = 2. We then apply this concavity property to obtain the following functional inequality (likewise extending the previously established result for the case that p = 2): For all r ∈ (0 , 1) , we have 2 β π p + 1 < β − log ⁡ (r ′) K p (r) + β − log ⁡ (r) K p (r ′) ≤ 2 β + 2 log ⁡ (2 p) K p (1 / 2 p) , where r ′ = 1 − r p p , p ∈ (1 , ∞) , and β ≥ λ (p). Both bounds are sharp. The sign of equality holds if and only if r = 1 / 2 p . [ABSTRACT FROM AUTHOR]

In this paper we study sharp estimates of pre-Schwarzian derivatives of functions belonging to the Nehari-type classes by using techniques from differential equations. In the sequel, we also see that a solution of a complex differential equation has a special form in terms of ratio of hypergeometric functions resulting to an integral representation. Finally, we attempt to study those univalent functions in the unit disk for which the image domain is an unbounded John domain. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to introduce a further extension of the extended Caputo fractional derivative operator and establish the extension of an extended fractional derivative of some known elementary functions. Additionally, we investigate the extended fractional derivative of some familiar special functions, the Mellin transform of the newly defined Caputo fractional derivative operator and generating relations for the extensions of extended hypergeometric functions. [ABSTRACT FROM AUTHOR]

This paper deals with the evaluation of some definite Euler-type integrals in terms of the Wright hypergeometric function. We obtain a theorem on the Wright hypergeometric function and then use this theorem to evaluate some definite integrals. Further, we derive some results as applications of these evaluations. Multi-variable cases of the derived results of this paper are also briefly discussed. [ABSTRACT FROM AUTHOR]