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A fractional power interpretation of the Laguerre derivative (D x D) α , D ≡ d / d x is discussed. The corresponding fractional integrals are introduced. Mapping and semigroup properties, integral representations and Mellin transform analysis are presented. A relationship with the Riemann–Liouville fractional integrals is demonstrated. Finally, a second kind integral equation of the Volterra type, involving the Laguerre fractional integral is solved in terms of the double hypergeometric type series as the resolvent kernel. [ABSTRACT FROM AUTHOR]

In this work we investigate the heat kernel of the Laplace–Beltrami operator on a rectangular torus and the according temperature distribution. We compute the minimum and the maximum of the temperature on rectangular tori of fixed area by means of Gauss' hypergeometric function 2 F 1 and the elliptic modulus. In order to be able to do this, we employ a beautiful result of Ramanujan, connecting hypergeometric functions, the elliptic modulus and theta functions. Also, we investigate the temperature distribution of the heat kernel on hexagonal tori and use Ramanujan's corresponding theory of signature 3 to derive analogous results to the rectangular case. Lastly, we show connections to the problem of finding the exact value of Landau's "Weltkonstante", a universal constant arising in the theory of extremal holomorphic mappings; and for a related, restricted extremal problem we show that the conjectured solution is the second lemniscate constant. [ABSTRACT FROM AUTHOR]

The present paper deals with a new different generalization of the Mittag-Leffler function through q-calculus. We then investigate its remarkable properties like convergence, recurrence relation, integral representation, q-derivative formula, q-Laplace transformation, and image formula under q-derivative operator. In addition to this, we consider some specific cases to give the utilization of our main results. [ABSTRACT FROM AUTHOR]

An integrating factor f ~ x is presented involving the terms in y ′ ′ x and q x y x of the general homogenous second-order linear ordinary differential equation. The new integrating factors obey second-order differential equations, rather than being given by quadrature. The new factors provides old and new integrals for special functions which obey such differential equations. The functions considered here are cylinder functions, parabolic cylinder functions and Whittaker functions. All the integrals given have been checked using Mathematica. [ABSTRACT FROM AUTHOR]

In this article, we define a convolution operator and study its boundedness on mixed-norm spaces. In particular, we obtain a well-known result on the boundedness of composition operators given by Avetisyan and Stević in [K. Avetisyan and S. Stević, The generalized Libera transform is bounded on the Besov mixed-norm, BMOA and VMOA spaces on the unit disc, Appl. Math. Comput. 213 2009, 2, 304–311]. Also we consider the adjoint 𝒜 b , c {\mathcal{A}^{b,c}} for b > 0 {b>0} of two parameter families of Cesáro averaging operators and prove the boundedness on Besov mixed-norm spaces B α + (c - 1) p , q {B_{\alpha+(c-1)}^{p,q}} for c > 1 {c>1}. [ABSTRACT FROM AUTHOR]

In this article starting from some reductions of hyperelliptic integrals of genus 3 into elliptic integrals, due to Michael Roberts (A Tract on the addition of Elliptic and hyperelliptic integrals, Hodger, Foster and Co, 1871) we obtain several identities which, to the best of our knowledge, are all new. The strategy used at this purpose is to evaluate Roberts integrals, in two different ways, on one side by means of elliptic integrals, obtained from the Roberts method of reduction and, on the other side, using multivariate hypergeometric functions. [ABSTRACT FROM AUTHOR]

Asymptotic expansions of the Gauss hypergeometric function with large parameters, $F(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D716}_{1}\unicode[STIX]{x1D70F},\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D716}_{2}\unicode[STIX]{x1D70F};\unicode[STIX]{x1D6FE}+\unicode[STIX]{x1D716}_{3}\unicode[STIX]{x1D70F};z)$ as $|\unicode[STIX]{x1D70F}|\rightarrow \infty$ , are known for many special cases, but not for one that the author encountered in recent work on fluid mechanics: $\unicode[STIX]{x1D716}_{2}=0$ and $\unicode[STIX]{x1D716}_{3}=\unicode[STIX]{x1D716}_{1}z$. This paper gives the leading term for that case if $\unicode[STIX]{x1D6FD}$ is not a negative integer and $z$ is not on the branch cut $[1,\infty)$ , and it shows how subsequent terms can be found. [ABSTRACT FROM AUTHOR]

We prove that the function G a (x) = a − log ⁡ (1 − x) K (x) (a ∈ R) is strictly concave on (0 , 1) if and only if a ≥ 8 / 5. This solves a problem posed by Yang and Tian and complements their result that 1 / G a (a ≥ 0) is strictly concave on (0 , 1) if and only if a = 4 / 3. Moreover, we apply our concavity theorem to present several functional inequalities involving K . Among others, we prove that if a ≥ 8 / 5 , then 2 a π + 1 < a − log ⁡ (r ′) K (r) + a − log ⁡ (r) K (r ′) ≤ 2 a + log ⁡ (2) K (1 / 2) for all r ∈ (0 , 1) , where r ′ = 1 − r 2 . Both bounds are sharp and the sign of equality holds if and only if r = 1 / 2 . [ABSTRACT FROM AUTHOR]

The problem of parameter estimation for the non-stationary ergodic diffusion with Fisher-Snedecor invariant distribution, to be called Fisher-Snedecor diffusion, is considered. We propose generalized method of moments (GMM) estimator of unknown parameter, based on continuous-time observations, and prove its consistency and asymptotic normality. The explicit form of the asymptotic covariance matrix in asymptotic normality framework is calculated according to the new iterative technique based on evolutionary equations for the point-wise covariations. The results are illustrated in a simulation study covering various starting distributions and parameter values. [ABSTRACT FROM AUTHOR]

We investigate the analytic properties of a new class of special functions that appear in the kernels of a class of integral operators underlying the dynamics of matter relaxation processes in attractive fields. These functions, recently introduced by the second author, generate the kernels of the principal parts of these operators and play an important role in understanding their spectral characteristics. We reveal the representations of these functions in terms of the Gauss and Clausen hypergeometric functions and present differential-difference and differential equations they satisfy. Mathematically, the results include calculation of certain trigonometric double integrals and derivation of their other properties. Furthermore, they represent a potentially useful tool in matter relaxation in an external field, the study of nanoelectronic electrolyte-based systems and dynamics of charge carriers in media with obstacles. [ABSTRACT FROM AUTHOR]

Discrete analogues of the classical Mehler–Fock transforms are introduced and investigated. It involves series with the associated Legendre function P i n − 1 / 2 μ (x) , x>1, R e μ < 1 / 2 , n ∈ N , i is the imaginary unit, and the so-called incomplete Legendre integrals. Several expansions of arbitrary functions and sequences in terms of these series and integrals are established. [ABSTRACT FROM AUTHOR]

In this article we find finite field analogues of certain product formulas satisfied by the classical hypergeometric series. We express product of two 2 F 1 -Gaussian hypergeometric series as 4 F 3 - and 3 F 2 -Gaussian hypergeometric series. We use properties of Gauss and Jacobi sums and our earlier works on finite field Appell series to deduce these product formulas satisfied by the Gaussian hypergeometric series. We then use these transformations to evaluate explicitly some special values of 4 F 3 - and 3 F 2 -Gaussian hypergeometric series. By counting points on CM elliptic curves over finite fields, Ono found certain special values of 2 F 1 - and 3 F 2 -Gaussian hypergeometric series containing trivial and quadratic characters as parameters. Later, Evans and Greene found special values of certain 3 F 2 -Gaussian hypergeometric series containing arbitrary characters as parameters from where some of the values obtained by Ono follow as special cases. We show that some of the results of Evans and Greene follow from our product formulas including a finite field analogue of the classical Clausen's identity. [ABSTRACT FROM AUTHOR]

In this paper, we contribute to the results of Bakhet et al. (Integral Transforms Spec. Funct. 30:138–156, 2019) by applying fractional operators to the Wright hypergeometric matrix functions. We give matrix recurrence relations and integral formulas for the Wright hypergeometric matrix functions. We also clarify particular cases of the main results. [ABSTRACT FROM AUTHOR]

This article aims to establish certain image formulas associated with the fractional calculus operators with Appell function in the kernel and Caputo-type fractional differential operators involving Srivastava polynomials and extended Mittag-Leffler function. The main outcomes are presented in terms of the extended Wright function. In addition, along with the noted outcomes, the implications are also highlighted. [ABSTRACT FROM AUTHOR]

Some general integrals are presented which were obtained from two integrating factors f (x) and f ˆ (x) for the first two and last two terms, respectively, of the second-order linear ordinary differential equation obeyed by many special functions. The integrating factor f (x) is identical to the Lagrangian factor introduced in Conway [A Lagrangian method for deriving new indefinite integrals of special functions. Integral Transforms Spec Funct. 2015;26(10):812–824; Indefinite integrals of some special functions from a new method. Integral Transforms Spec Funct. 2015;26(11):845–858] and sometimes provides textbook integrals. The integrals provided by the second factor seem to have been little explored. Results are given for Gauss hypergeometric functions and Kummer confluent hypergeometric functions. All results presented have been checked using Mathematica. [ABSTRACT FROM AUTHOR]

In this paper, the authors present sharp approximations in terms of sine function and polynomials for the so-called Ramanujan constant (or the Ramanujan R-function) R(a), by showing some monotonicity, concavity and convexity properties of certain combinations defined in terms of R(a), sin (π a) and polynomials. Some properties of the Riemann zeta function and its related special sums are presented, too. [ABSTRACT FROM AUTHOR]

In this paper, we prove six Ramanujan's modular equations of septic degree by employing Ramanujan's 1 ψ 1 summation formula and certain theta function identities. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to introduce a class of new extended forms of the beta function, Gauss hypergeometric function and Appell-Lauricella hypergeometric functions by means of the modified Bessel function of the third kind. Some typical generating relations for these extended hypergeometric functions are obtained by defining the extension of the Riemann-Liouville fractional derivative operator. Their connections with elementary functions and Fox’s H-function are also presented.

The method of brackets is an efficient method for the evaluation of alarge class of definite integrals on the half-line. It is based on a small collection of rules, some of which are heuristic. The extension discussed here is based on the concepts of null and divergent series. These are formal representations of functions, whose coefficients an have meromorphic representations for n ∈ ℂ, but might vanish or blow up when n ∈ ℕ. These ideas are illustrated with the evaluation of a variety of entries from the classical table of integrals by Gradshteyn and Ryzhik.

In the article, we present several new monotonicity properties and bounds involving the generalized Grötzsch ring functions μ a , b in the theory of Ramanujan's generalized modular equation for 0 < a , b < 1 . Our results are the variants and extensions of some previously known results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce some new polynomials obtained from second- and third-order algebraic equations by using a simple iterative method. One-variable polynomials obtained in this study deal with special form of Pöschl–Teller potential with constant energy, and the two-variable polynomials are related to time-dependent wave equations. We present some recurrence relations, Binet formula and get various families of linear, multilinear and multilateral generating functions for these polynomials. In addition, we derive some special cases. At the end of the paper we also give an extension to the multidimensional case of our results. [ABSTRACT FROM AUTHOR]

New relations for the first Appell hypergeometric matrix function F 1 (A , B , B ′ , C ; z , w) are obtained including generating matrix functions, contiguous relations, recursion formulas, differentiations and series. [ABSTRACT FROM AUTHOR]

In recent years, special matrix functions and polynomials of a real or complex variable have been in a focus of increasing attention leading to new and interesting problems. In this work, we present matrix space analogues to generalized some functions and polynomials in the framework of matrix setting. Many of the special matrix functions and polynomials are constructed along standard procedures. Recently published papers are also surveyed and we list the most essential ones. [ABSTRACT FROM AUTHOR]

We consider an extension of the Ramanujan series with a variable $x$. If we let $x=x_{0}$ , we call the resulting series 'Ramanujan series with the shift $x_{0}$ '. Then we relate these shifted series to some $q$ -series and solve the case of level $4$ with the shift $x_{0}=1/2$. Finally, we indicate a possible way towards proving some patterns observed by the author corresponding to the levels $\ell =1,2,3$ and the shift $x_{0}=1/2$. [ABSTRACT FROM AUTHOR]

We prove Laporta's conjecture which relates the 4-loop sunrise diagram in 2-dimensional quantum field theory to Watson's integral for 4-dimensional hypercubic lattice. We also establish several related integral identities proposed by Laporta, including a reduction of the 4-loop sunrise diagram to special values of Euler's gamma function and generalized hypergeometric series: 4 π 5 / 2 3 3 2 6 Γ 1 3 π 9 4 F 3 1 6 , 1 3 , 1 3 , 1 2 2 3 , 5 6 , 5 6 1 - 2 4 3 π Γ 1 3 9 4 F 3 1 2 , 2 3 , 2 3 , 5 6 7 6 , 7 6 , 4 3 1. [ABSTRACT FROM AUTHOR]

Positive definite matrix functions on spheres arise naturally in multivariate approximation and spatial statistics. The construction of strictly positive definite models has become one of the most important goals for the analysis and study of vector valued random fields on spheres. This paper derives strictly positive definite models based on hypergeometric functions. In statistical nomenclature, it describes strictly positive definite covariance functions of stationary, isotropic and mean square continuous Gaussian vector random fields on spheres in which the isotropic part of the covariance functions are set through hypergeometric functions. [ABSTRACT FROM AUTHOR]

In this paper various analytic techniques are combined in order to study the average of a product of a Hecke L ‐function and a symmetric square L ‐function at the central point in the weight aspect. The evaluation of the second main term relies on the theory of Maaß forms of half‐integral weight and the Rankin–Selberg method. The error terms are bounded using the Liouville–Green approximation. [ABSTRACT FROM AUTHOR]

In the article, we present several monotonicity theorems and inequalities for the modular equation functions m a (r) and μ a (r) , and find the infinite-series formulas for m 1 / 3 (r) and m 1 / 4 (r) which depend only on r. As applications, we find several precise explicit estimates for the solution of Ramanujan's generalized modular equation. [ABSTRACT FROM AUTHOR]

Motivated by the study of symmetry breaking operators for indefinite orthogonal groups, we give a Gegenbauer expansion of the two variable function in terms of the ultraspherical polynomials and. Generalization, specialization, and limits of the expansion are also discussed. [ABSTRACT FROM AUTHOR]

Motivated by the study of symmetry breaking operators for indefinite orthogonal groups, we give a Gegenbauer expansion of the two variable function in terms of the ultraspherical polynomials and. Generalization, specialization, and limits of the expansion are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we show formally how can we generate continued fractions starting from three-term contiguous relations. Then as examples, starting from certain contiguous relations for 2 F 1 and 2 ϕ 1 , we derive orthogonal polynomials, continued fractions and in case of 2 F 1 , the measure of orthogonality. Also, a generalization of q-Kummer's identity is introduced. [ABSTRACT FROM AUTHOR]

In this paper, we give the recursion formula (in a more general case) for the linearization coefficients (β l (N k) (A k)) for Bessel polynomials (q n) , mentioned by Berg and Vignat p. 21. (14) (Constr Approx 27:15–32, 2008), in the expansion q n 1 (a 1 u) q n 2 (a 2 u) ... q n k (a k u) = ∑ l = min (n 1 , n 2 , ... , n k) L k β l (N k) (A k) q l (u) , where u ∈ R , N k = (n 1 , n 2 , ... , n k) ∈ N k , A k = (a 1 , a 2 , ... , a k) ∈ R + k with ∑ 1 ≤ i ≤ k a i = 1 and L k = ∑ 1 ≤ i ≤ k n i . This recursion formula yields, again, the positivity of the coefficients. In addition, in the case when k = 3 , we give either an explicit formula for β l (N 3) (A 3) and we derive a double integral formula via convolution of Student t-densities. As a bonus, two reductions for this double integral are given; the first reduction uses an integral formula given by Atia and Zeng (Ramanujan J 28:211–221, 2012. 10.1007/s11139-011-9348-4), and the second uses the one given by Boros and Moll (J Comput Appl Math 106:361–368, 1999). [ABSTRACT FROM AUTHOR]