On the Asymptotic Expansions of the (<i>p</i>,<i>k</i>)-Analogues of the Gamma Function and Associated Functions
General asymptotic expansion of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow&g...
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MDPI AG
2025-01-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/14/1/55 |
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| author | Tomislav Burić |
| author_facet | Tomislav Burić |
| author_sort | Tomislav Burić |
| collection | DOAJ |
| description | General asymptotic expansion of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula>-gamma function is obtained and various approaches to this expansion are studied. The numerical precision of the derived asymptotic formulas is shown and compared. Results are applied to the analogues of digamma and polygamma functions, and asymptotic expansion of the quotient of two <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula>-gamma functions is also derived and analyzed. Various examples and application to the <i>k</i>-Pochhammer symbol are presented. |
| format | Article |
| id | doaj-art-afa8d187e9224627aed6e2f2a04c9867 |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-afa8d187e9224627aed6e2f2a04c98672025-01-24T13:22:17ZengMDPI AGAxioms2075-16802025-01-011415510.3390/axioms14010055On the Asymptotic Expansions of the (<i>p</i>,<i>k</i>)-Analogues of the Gamma Function and Associated FunctionsTomislav Burić0Faculty of Electrical Engineering and Computing, University of Zagreb, Unska 3, 10000 Zagreb, CroatiaGeneral asymptotic expansion of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula>-gamma function is obtained and various approaches to this expansion are studied. The numerical precision of the derived asymptotic formulas is shown and compared. Results are applied to the analogues of digamma and polygamma functions, and asymptotic expansion of the quotient of two <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula>-gamma functions is also derived and analyzed. Various examples and application to the <i>k</i>-Pochhammer symbol are presented.https://www.mdpi.com/2075-1680/14/1/55asymptotic expansiongamma function(<i>p</i>,<i>k</i>)-gamma functionquotient of gamma functions<i>k</i>-Pochhammer symbol |
| spellingShingle | Tomislav Burić On the Asymptotic Expansions of the (<i>p</i>,<i>k</i>)-Analogues of the Gamma Function and Associated Functions Axioms asymptotic expansion gamma function (<i>p</i>,<i>k</i>)-gamma function quotient of gamma functions <i>k</i>-Pochhammer symbol |
| title | On the Asymptotic Expansions of the (<i>p</i>,<i>k</i>)-Analogues of the Gamma Function and Associated Functions |
| title_full | On the Asymptotic Expansions of the (<i>p</i>,<i>k</i>)-Analogues of the Gamma Function and Associated Functions |
| title_fullStr | On the Asymptotic Expansions of the (<i>p</i>,<i>k</i>)-Analogues of the Gamma Function and Associated Functions |
| title_full_unstemmed | On the Asymptotic Expansions of the (<i>p</i>,<i>k</i>)-Analogues of the Gamma Function and Associated Functions |
| title_short | On the Asymptotic Expansions of the (<i>p</i>,<i>k</i>)-Analogues of the Gamma Function and Associated Functions |
| title_sort | on the asymptotic expansions of the i p i i k i analogues of the gamma function and associated functions |
| topic | asymptotic expansion gamma function (<i>p</i>,<i>k</i>)-gamma function quotient of gamma functions <i>k</i>-Pochhammer symbol |
| url | https://www.mdpi.com/2075-1680/14/1/55 |
| work_keys_str_mv | AT tomislavburic ontheasymptoticexpansionsoftheipiikianaloguesofthegammafunctionandassociatedfunctions |