arXiv subjects

Dmitrii Karp

Publications and source records attributed to Dmitrii Karp.

27 records · Page 2Linked to original sources

arXiv2020

Completely monotonic ratios of basic and ordinary gamma functions

We investigate conditions for logarithmic complete monotonicity of product ratios of gamma and q-gamma functions whose arguments are linear functions of the variable. We give necessary and sufficient conditions in terms of nonnegativity of a certain explicitly written measure in the q case and of a certain elementary function in the classical q=1 case. In the latter case we further provide simple new sufficient conditions leading to many new examples of logarithmically completely monotonic gamma ratios. Finally, we apply some of our results to study monotonicity of some gamma ratios and rational functions.

math.CA↗

arXiv2019

The Fox-Wright function near the singularity and branch cut

The Fox-Wright function is a further extension of the generalized hypergeometric function obtained by introducing arbitrary positive scaling factors into the arguments of the gamma functions in the summand. Its importance comes mostly from its role in fractional calculus although other interesting applications also exist. If the sums of the scaling factors in the top and bottom parameters are equal, the series defining the Fox-Wright function has a finite non-zero radius of convergence. It was demonstrated by Braaksma in 1964 that the Fox-Wright function can then be extended to a holomorphic function in the complex plane cut along a ray from the positive point on the boundary of the disk of convergence to the point at infinity. In this paper we study the behavior of the Fox-Wright function in the neighborhood of this positive singular point. Under certain restrictions we give a convergent expansion with recursively computed coefficients completely characterizing this behavior. We further compute the jump and the average value of the Fox-Wright function on the banks of the branch cut.

math.CA↗

arXiv2019

A new identity for a sum of products of the generalized hypergeometric functions

Reduction formulas for sums of products of hypergeometric functions can be traced back to Euler. This topic has an intimate connection to summation and transformation formulas, contiguous relations and algebraic properties of the (generalized) hypergeometric differential equation. Over recent several years, important discoveries have been made in this subject by Gorelov, Ebisu, Beukers and Jouhet and Feng, Kuznetsov and Yang. In this paper, we give a generalization of Feng, Kuznetsov and Yang identity covering also Ebisu's and Gorelov's formulas as particular cases.

math.CA↗

arXiv2019

A note on Fox's H function in the light of Braaksma's results

In our previous works we found a power series expansion of a particular case of Fox's $H$ function $H^{q,0}_{p,q}$ in a neighborhood of its positive singularity. An inverse factorial series expansion of the integrand of $H^{q,0}_{p,q}$ served as our main tool. However, a necessary restriction on parameters is missing in those works. In this note we fill this gap and give a simpler and shorter proof of the expansion around the positive singular point. We further identify more precisely the abscissa of convergence of the underlying inverse factorial series. Our new proof hinges on a slight generalization of a particular case of Braaksma's theorem about analytic continuation of Fox's $H$ function.

math.CV↗

arXiv2016

Hypergeometric Differential Equation and New Identities for the Coefficients of Nørlund and Bühring

The fundamental set of solutions of the generalized hypergeometric differential equation in the neighborhood of unity has been built by Nørlund in 1955. The behavior of the generalized hypergeometric function in the neighborhood of unity has been described in the beginning of 1990s by Bühring, Srivastava and Saigo. In the first part of this paper we review their results rewriting them in terms of Meijer's $G$-function and explaining the interconnections between them. In the second part we present new formulas and identities for the coefficients that appear in the expansions of Meijer's $G$-function and generalized hypergeometric function around unity. Particular cases of these identities include known and new relations for Thomae's hypergeometric function and forgotten Hermite's identity for the sine function.

math.CA↗

arXiv2015

Normalized incomplete beta function: log-concavity in parameters and other properties

The normalized incomplete beta function can be defined either as cumulative distribution function of beta density or as the Gauss hypergeometric function with one of the upper parameters equal to unity. Logarithmic concavity/convexity of this function in parameters was established by Finner and Roters in 1997. Their proof is indirect and rather difficult; it is based on generalized reproductive property of certain more general distributions. These authors remark that these results "seems to be very hard to obtain by usual analytic methods". In the first part of this paper we provide such proof based on standard tools of analysis. In the second part we go one step further and investigate the sign of generalized Turán determinants formed by shifts of the normalized incomplete beta function. Under some additional restrictions we demonstrate that these coefficients are of the same sign. We further conjecture that such restrictions can be removed without altering the results. Our method of proof also leads to various companion results which may be of independent interest. In particular, we establish linearization formulas and two-sided bounds for the above mentioned Turán determinants. Further, we find two combinatorial style identities for finite sums which we believe to be new.

math.CA↗

arXiv2015

Completely monotonic gamma ratio and infinitely divisible H-function of Fox

We investigate conditions for logarithmic complete monotonicity of a quotient of two products of gamma functions, where the argument of each gamma function has different scaling factor. We give necessary and sufficient conditions in terms of nonnegativity of some elementary function and more practical sufficient conditions in terms of parameters. Further, we study the representing measure in Bernstein's theorem for both equal and non-equal scaling factors. This leads to conditions on parameters under which Meijer's $G$-function or Fox's $H$-function represents an infinitely divisible probability distribution on positive half-line. Moreover, we present new integral equations for both $G$-function and $H$-function. The results of the paper generalize those due to Ismail (with Bustoz, Muldoon and Grinshpan) and Alzer who considered previously the case of unit scaling factors.

math.CA↗

arXiv2014

Representations and inequalities for generalized hypergeometric functions

We find an integral representation for the generalized hypergeometric function unifying known representations via generalized Stieltjes, Laplace and cosine Fourier transforms. Using positivity conditions for the weight in this representation we establish various new facts regarding generalized hypergeometric functions, including complete monotonicity, log-convexity in upper parameters, monotonicity of ratios and new proofs of Luke's bounds. Besides, we derive two-sided inequalities for the Bessel type hypergeometric functions by using their series representations.

math.CA↗

arXiv2012

The Multivariate Watson Distribution: Maximum-Likelihood Estimation and other Aspects

This paper studies fundamental aspects of modelling data using multivariate Watson distributions. Although these distributions are natural for modelling axially symmetric data (i.e., unit vectors where $\pm \x$ are equivalent), for high-dimensions using them can be difficult. Why so? Largely because for Watson distributions even basic tasks such as maximum-likelihood are numerically challenging. To tackle the numerical difficulties some approximations have been derived---but these are either grossly inaccurate in high-dimensions (\emph{Directional Statistics}, Mardia & Jupp. 2000) or when reasonably accurate (\emph{J. Machine Learning Research, W. & C.P., v2}, Bijral \emph{et al.}, 2007, pp. 35--42), they lack theoretical justification. We derive new approximations to the maximum-likelihood estimates; our approximations are theoretically well-defined, numerically accurate, and easy to compute. We build on our parameter estimation and discuss mixture-modelling with Watson distributions; here we uncover a hitherto unknown connection to the "diametrical clustering" algorithm of Dhillon \emph{et al.} (\emph{Bioinformatics}, 19(13), 2003, pp. 1612--1619).

stat.CO↗