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J. L. González-Santander

Publications and source records attributed to J. L. González-Santander.

8 recordsLinked to original sources
arXiv2025

On the application of a hypergeometric identity to generate generalized hypergeometric reduction formulas

We systematically exploit a new generalized hypergeometric identity to obtain new hypergeometric summation formulas. As a consistency test, alternative proofs for some special cases are also provided. As a byproduct new summation formulas involving finite sums involving the psi function, or a recursive formula for the Bateman's G function are derived. Finally, all the results have been numerically checked with MATHEMATICA.

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arXiv2022

A note on some identities involving special functions from the hypergeometric solution of algebraic equations

From the algebraic solution of $x^{n}-x+t=0$ for $n=2,3,4$ and the corresponding solution in terms of hypergeometric functions, we obtain a set of reduction formulas for hypergeometric functions. By differentiation and integration of these results, and applying other known reduction formulas of hypergeometric functions, we derive new reduction formulas of special functions as well as the calculation of some infinite integrals in terms of elementary functions.

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arXiv2018

Reflection formulas for order derivatives of Bessel functions

From new integral representations of the $n$-th derivative of Bessel functions with respect to the order, we derive some reflection formulas for the first and second order derivative of $J_{ν}\left( t\right) $ and $% Y_{ν}\left( t\right) $ for integral order, and for the $n$-th order derivative of $I_{ν}\left( t\right) $ and $K_{ν}\left( t\right) $ for arbitrary real order. As an application of the reflection formulas obtained for the first order derivative, we extend some formulas given in the literature to negative integral order. Also, as a by-product, we calculate an integral which does not seem to be reported in the literature.

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arXiv2018

On the $n$-th derivative and the fractional integration of Bessel functions with respect to the order

We obtain integral representations of the $n$-th derivatives of the Bessel functions with respect to the order. The numerical evaluation of these expressions is very efficient using a double exponential integration strategy. Also, from the integral representation corresponding to the Macdonald function, we have calculated a new integral. Finally, we calculate integral expressions for the fractional derivatives of the Bessel functions with respect to the order. Simple proofs for some particular cases given in the literature are provided as well.

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arXiv2018

A note on the order derivatives of Kelvin functions

We calculate the derivative of the $\mathrm{ber}_{ν}$, $\,\mathrm{bei}_{ν}$, $\mathrm{ker}_{ν}$, and $\,\mathrm{kei}_{ν}$ functions with respect to the order $ν$ in closed-form for $ν\in \mathbb{R}$. Unlike the expressions found in the literature for order derivatives of the $\mathrm{ber}_{ν}$ and $\,\mathrm{bei}_{ν}$ functions, we provide much more simple expressions that are also applicable for negative integral order. The expressions for the order derivatives of the $\mathrm{ker}_{ν}$ and $\,\mathrm{kei}_{ν}$ functions seem to be novel. Also, as a by-product, we calculate some new integrals involving the $\mathrm{ber}_{ν}$ and $\,\mathrm{bei}_{ν}$ functions in closed-form. Finally, we include a simple derivation of some integral representations of the $\mathrm{ber}$ and $\,\mathrm{bei}$ functions.

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arXiv2018

Closed form expressions for derivatives of Bessel functions with respect to the order

We have used recent integral representations of the derivatives of the Bessel functions with respect to the order to obtain closed-form expressions in terms of generalized hypergeometric functions and Meijer-$G$ functions. Also, we have carried out similar calculations for the derivatives of the modified Bessel functions with respect to the order, obtaining closed-form expressions as well. For this purpose, we have obtained integral representations of the derivatives of the modified Bessel functions with respect to the order. As by-products, we have calculated two non-tabulated integrals.

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arXiv2018

Recursive formulas for $_{2}F_{1}$ and $_{3}F_{2}$ hypergeometric series

Recursive formulas extending some known $_{2}F_{1}$ and $_{3}F_{2}$ summation formulas by using contiguous relations have been obtained. On the one hand, these recursive equations are quite suitable for symbolic and numerical evaluation by means of computer algebra. On the other hand, sometimes closed-forms of such extensions can be derived by induction. It is expected that the method used to obtain the different recursive equations can be applied to extend other hypergeometric summation formulas given in the literature.

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