arXiv subjects

S. M. Sitnik

Publications and source records attributed to S. M. Sitnik.

21 records · Page 2Linked to original sources

arXiv2008

Inequalities and monotonicity of ratios for generalized hypergeometric function

We find two-sided inequalities for the generalized hypergeometric function of the form ${_{q+1}}F_{q}(-x)$ with positive parameters restricted by certain additional conditions. Both lower and upper bounds agree with the value of ${_{q+1}}F_{q}(-x)$ at the endpoints of positive semi-axis and are asymptotically precise at one of the endpoints. The inequalities are derived from a theorem asserting the monotony of the quotient of two generalized hypergeometric functions with shifted parameters. The proofs hinge on a generalized Stieltjes representation of the generalized hypergeometric function. This representation also provides yet another method to deduce the second Thomae relation for ${_{3}F_{2}}(1)$ and leads to an integral representations of ${_{4}F_{3}}(x)$ in terms of the Appell function $F_3$. In the last section of the paper we list some open questions and conjectures.

math.CA↗

arXiv2006

Asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity

We find two convergent series expansions for Legendre's first incomplete elliptic integral $F(λ,k)$ in terms of recursively computed elementary functions. Both expansions are valid at every point of the unit square $0<λ,k<1$. Truncated expansions yield asymptotic approximations for $F(λ,k)$ as $λ$ and/or $k$ tend to unity, including the case when logarithmic singularity $λ=k=1$ is approached from any direction. Explicit error bounds are given at every order of approximation. For the reader's convenience we present explicit expressions for low-order approximations and numerical examples to illustrate their accuracy. Our derivation is based on rearrangements of some known double series expansions, hypergeometric summation algorithms and inequalities for hypergeometric functions.

math.CA↗

arXiv2006

Series expansions for the third incomplete elliptic integral via partial fraction decompositions

We find convergent double series expansions for Legendre's third incomplete elliptic integral valid in overlapping subdomains of the unit square. Truncated expansions provide asymptotic approximations in the neighbourhood of the logarithmic singularity $(1,1)$ if one of the variables approaches this point faster than the other. Each approximation is accompanied by an error bound. For a curve with an arbitrary slope at $(1,1)$ our expansions can be rearranged into asymptotic expansions depending on a point on the curve. For reader's convenience we give some numeric examples and explicit expressions for low-order approximations.

math.CA↗