Sums Involving the Digamma Function Connected to the Incomplete Beta Function and the Bessel functions
We calculate some infinite sums containing the digamma function in closed form. These sums are related either to the incomplete beta function or to the Bessel functions. The calculations yield interesting new results as by-products, such as parameter differentiation formulas for the beta incomplete...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-04-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/8/1937 |
| _version_ | 1797604439792549888 |
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| author | Juan Luis González-Santander Fernando Sánchez Lasheras |
| author_facet | Juan Luis González-Santander Fernando Sánchez Lasheras |
| author_sort | Juan Luis González-Santander |
| collection | DOAJ |
| description | We calculate some infinite sums containing the digamma function in closed form. These sums are related either to the incomplete beta function or to the Bessel functions. The calculations yield interesting new results as by-products, such as parameter differentiation formulas for the beta incomplete function, reduction formulas of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mn>3</mn></msub><msub><mi>F</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula> hypergeometric functions, or a definite integral which does not seem to be tabulated in the most common literature. As an application of certain sums involving the digamma function, we calculated some reduction formulas for the parameter differentiation of the Mittag–Leffler function and the Wright function. |
| first_indexed | 2024-03-11T04:46:36Z |
| format | Article |
| id | doaj.art-27fc52514a204668b27583627bc20e14 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T04:46:36Z |
| publishDate | 2023-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-27fc52514a204668b27583627bc20e142023-11-17T20:18:44ZengMDPI AGMathematics2227-73902023-04-01118193710.3390/math11081937Sums Involving the Digamma Function Connected to the Incomplete Beta Function and the Bessel functionsJuan Luis González-Santander0Fernando Sánchez Lasheras1Department of Mathematics, Universidad de Oviedo, 33007 Oviedo, Asturias, SpainDepartment of Mathematics, Universidad de Oviedo, 33007 Oviedo, Asturias, SpainWe calculate some infinite sums containing the digamma function in closed form. These sums are related either to the incomplete beta function or to the Bessel functions. The calculations yield interesting new results as by-products, such as parameter differentiation formulas for the beta incomplete function, reduction formulas of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mn>3</mn></msub><msub><mi>F</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula> hypergeometric functions, or a definite integral which does not seem to be tabulated in the most common literature. As an application of certain sums involving the digamma function, we calculated some reduction formulas for the parameter differentiation of the Mittag–Leffler function and the Wright function.https://www.mdpi.com/2227-7390/11/8/1937digamma functionBessel functionsincomplete beta functionWright functionMittag–Leffler functiondifferentiation with respect to parameters |
| spellingShingle | Juan Luis González-Santander Fernando Sánchez Lasheras Sums Involving the Digamma Function Connected to the Incomplete Beta Function and the Bessel functions Mathematics digamma function Bessel functions incomplete beta function Wright function Mittag–Leffler function differentiation with respect to parameters |
| title | Sums Involving the Digamma Function Connected to the Incomplete Beta Function and the Bessel functions |
| title_full | Sums Involving the Digamma Function Connected to the Incomplete Beta Function and the Bessel functions |
| title_fullStr | Sums Involving the Digamma Function Connected to the Incomplete Beta Function and the Bessel functions |
| title_full_unstemmed | Sums Involving the Digamma Function Connected to the Incomplete Beta Function and the Bessel functions |
| title_short | Sums Involving the Digamma Function Connected to the Incomplete Beta Function and the Bessel functions |
| title_sort | sums involving the digamma function connected to the incomplete beta function and the bessel functions |
| topic | digamma function Bessel functions incomplete beta function Wright function Mittag–Leffler function differentiation with respect to parameters |
| url | https://www.mdpi.com/2227-7390/11/8/1937 |
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