An approximation for zero-balanced Appell function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mi>F</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math> near <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>

We find two series expansions for Legendre’s second incomplete elliptic integral E ( λ , k ) E(\lambda , k) in terms of recursively computed elementary functions. Both expansions converge at every point of the unit square in the ( λ , k ) (\lambda , k) plane. Partial sums of the proposed expansions form a sequence of approximations to E ( λ , k ) E(\lambda ,k) which are asymptotic when λ \lambda and/or k k tend to unity, including when both approach the logarithmic singularity λ = k = 1 \lambda =k=1 from any direction. Explicit two-sided error bounds are given at each approximation order. These bounds yield a sequence of increasingly precise asymptotically correct two-sided inequalities for E ( λ , k ) E(\lambda , k) . For the reader’s convenience we further present explicit expressions for low-order approximations and numerical examples to illustrate their accuracy. Our derivations are based on series rearrangements, hypergeometric summation algorithms and extensive use of the properties of the generalized hypergeometric functions including some recent inequalities.

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Some zero-balanced terminating hypergeometric series and their applications

Srivastava,

Malik,

Qureshi

et al. 2023

Filomat

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Various families of such Special Functions as the hypergeometric functions of one, two and more variables, and their associated summation, transformation and reduction formulas, are potentially useful not only as solutions of ordinary and partial differential equations, but also in the widespread problems in the mathematical, physical, engineering and statistical sciences. The main object of this paper is first to establish four general double-series identities, which involve some suitably-bounded sequences of complex numbers, by using zero-balanced terminating hypergeometric summation theorems for the generalized hypergeometric series r+1Fr(1) (r = 1, 2, 3) in conjunction with the series rearrangement technique. The sum (or difference) of two general double hypergeometric functions of the Kamp? de F?riet type are then obtained in terms of a generalized hypergeometric function under appropriate convergence conditions. A closed form of the following Clausen hypergeometric function: 3F2 (?27z/4(1?z)3) and a reduction formula for the Srivastava-Daoust double hypergeometric function with the arguments (z,?z/4 ) are also derived. Many of the reduction formulas, which are established in this paper, are verified by using the software program, Mathematica. Some potential directions for further researches along the lines of this paper are also indicated.

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An approximation for zero-balanced Appell function F1 near (1,1)

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References 15 publications

Monotonicity of asymptotic relations for generalized hypergeometric functions

Monotonicity of asymptotic relations for generalized hypergeometric functions

Convergent expansions and bounds for the incomplete elliptic integral of the second kind near the logarithmic singularity

Some zero-balanced terminating hypergeometric series and their applications

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