A very general beam solution of the paraxial wave equation in circular cylindrical coordinates is presented. We call such a field a circular beam (CiB). The complex amplitude of the CiB is described by either the Whittaker functions or the confluent hypergeometric functions and is characterized by three parameters that are complex in the most general situation. The propagation through complex A...

tions F(3), F(4), Lauricella’s quadruple hypergeometric function F A , Exton’s multiple hypergeometric functions X E:G;H , K10, K13, X8, (k)H 2 , (k)H 4 , Erdélyi’s multiple hypergeometric function Hn,k, Khan and Pathan’s triple hypergeometric function H (P) 4 , Kampé de Fériet’s double hypergeometric function F E:G;H , Appell’s double hypergeometric function of the second kind F2, and the Sriv...

In this paper, we derive a necessary and sufficient condition on the parameters of the Hypergeometric distribution for weak convergence to a Normal limit. We establish a Berry-Esseen theorem for the Hypergeometric distribution solely under this necessary and sufficient condition. We further derive a nonuniform Berry-Esseen bound where the tails of the difference between the Hypergeometric and t...

In this paper, the distributions of the largest and smallest eigenvalues of complex Wishart matrices and the condition number of complex Gaussian random matrices are derived. These distributions are represented by complex hypergeometric functions of matrix arguments, which can be expressed in terms of complex zonal polynomials. Several results are derived on complex hypergeometric functions and...

Let A be the class of analytic functions in the open unit disk U . For complex numbers a, b and c (c 6= 0,−1,−2, ....), let I c be the operator defined on A by (I c (f))(z) = z2F1(a, b; c; z) ∗ f(z) where 2F1(a, b; c; z) is the Gaussian hypergeometric function. The function f in A is said to be in the class k − SP c if I a,b c (f) is a k-parabolic starlike function. For this class the Fekete-Sz...

We study the log-convexity of the extended beta functions. As a consequence, we establish Turán-type inequalities. The monotonicity, log-convexity, log-concavity of extended hypergeometric functions are deduced by using the inequalities on extended beta functions. The particular cases of those results also give the Turán-type inequalities for extended confluent and extended Gaussian hypergeomet...

Based on the Hadamard product of power series, polynomial series expansions for confluent hypergeometric functions M(a, c; ·) and for Gaussian hypergeometric functions F (a, b; c; ·) are introduced and studied. It turns out that the partial sums provide an interesting alternative for the numerical evaluation of the functions M(a, c; ·) and F (a, b; c; ·), in particular, if the parameters are al...

Exton introduced 20 distinct triple hypergeometric functions whose names are Xi (i = 1, . . . , 20) to investigate their twenty Laplace integral representations whose kernels include the confluent hypergeometric functions 0F1, 1F1, a Humbert function Ψ1, and a Humbert function Φ2. The object of this paper is to present 18 new integral representations of Euler type for the Exton hypergeometric f...

We derive an expression for the value ζQ(3) of the spectral zeta function ζQ(s) studied in [10, 11] for the non-commutative harmonic oscillator defined in [17] using a Gaussian hypergeometric function. In this study, two sequences of rational numbers, denoted J̃2(n) and J̃3(n), which can be regarded as analogues of the Apéry numbers, naturally arise and play a key role in obtaining the expression...