A General Family of <i>q</i>-Hypergeometric Polynomials and Associated Generating Functions
Basic (or <i>q</i>-) series and basic (or <i>q</i>-) polynomials, especially the basic (or <i>q</i>-) hypergeometric functions and the basic (or <i>q</i>-) hypergeometric polynomials are studied extensively and widely due mainly to their potential for...
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MDPI AG
2021-05-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/11/1161 |
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| author | Hari Mohan Srivastava Sama Arjika |
| author_facet | Hari Mohan Srivastava Sama Arjika |
| author_sort | Hari Mohan Srivastava |
| collection | DOAJ |
| description | Basic (or <i>q</i>-) series and basic (or <i>q</i>-) polynomials, especially the basic (or <i>q</i>-) hypergeometric functions and the basic (or <i>q</i>-) hypergeometric polynomials are studied extensively and widely due mainly to their potential for applications in many areas of mathematical and physical sciences. Here, in this paper, we introduce a general family of <i>q</i>-hypergeometric polynomials and investigate several <i>q</i>-series identities such as an extended generating function and a Srivastava-Agarwal type bilinear generating function for this family of <i>q</i>-hypergeometric polynomials. We give a transformational identity involving generating functions for the generalized <i>q</i>-hypergeometric polynomials which we have introduced here. We also point out relevant connections of the various <i>q</i>-results, which we investigate here, with those in several related earlier works on this subject. We conclude this paper by remarking that it will be a rather trivial and inconsequential exercise to give the so-called <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-variations of the <i>q</i>-results, which we have investigated here, because the additional parameter <i>p</i> is obviously redundant. |
| first_indexed | 2024-03-10T11:10:55Z |
| format | Article |
| id | doaj.art-0b1fad717fed48b78b6a909ab8b5bc2a |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T11:10:55Z |
| publishDate | 2021-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-0b1fad717fed48b78b6a909ab8b5bc2a2023-11-21T20:43:59ZengMDPI AGMathematics2227-73902021-05-01911116110.3390/math9111161A General Family of <i>q</i>-Hypergeometric Polynomials and Associated Generating FunctionsHari Mohan Srivastava0Sama Arjika1Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, CanadaDepartment of Mathematics and Informatics, University of Agadez, P.O. Box 199, Agadez 8000, NigerBasic (or <i>q</i>-) series and basic (or <i>q</i>-) polynomials, especially the basic (or <i>q</i>-) hypergeometric functions and the basic (or <i>q</i>-) hypergeometric polynomials are studied extensively and widely due mainly to their potential for applications in many areas of mathematical and physical sciences. Here, in this paper, we introduce a general family of <i>q</i>-hypergeometric polynomials and investigate several <i>q</i>-series identities such as an extended generating function and a Srivastava-Agarwal type bilinear generating function for this family of <i>q</i>-hypergeometric polynomials. We give a transformational identity involving generating functions for the generalized <i>q</i>-hypergeometric polynomials which we have introduced here. We also point out relevant connections of the various <i>q</i>-results, which we investigate here, with those in several related earlier works on this subject. We conclude this paper by remarking that it will be a rather trivial and inconsequential exercise to give the so-called <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-variations of the <i>q</i>-results, which we have investigated here, because the additional parameter <i>p</i> is obviously redundant.https://www.mdpi.com/2227-7390/9/11/1161basic (or <i>q</i>-) hypergeometric serieshomogeneous <i>q</i>-difference operator<i>q</i>-binomial theoremcauchy polynomialsAl-Salam-Carlitz <i>q</i>-polynomialsrogers type formulas |
| spellingShingle | Hari Mohan Srivastava Sama Arjika A General Family of <i>q</i>-Hypergeometric Polynomials and Associated Generating Functions Mathematics basic (or <i>q</i>-) hypergeometric series homogeneous <i>q</i>-difference operator <i>q</i>-binomial theorem cauchy polynomials Al-Salam-Carlitz <i>q</i>-polynomials rogers type formulas |
| title | A General Family of <i>q</i>-Hypergeometric Polynomials and Associated Generating Functions |
| title_full | A General Family of <i>q</i>-Hypergeometric Polynomials and Associated Generating Functions |
| title_fullStr | A General Family of <i>q</i>-Hypergeometric Polynomials and Associated Generating Functions |
| title_full_unstemmed | A General Family of <i>q</i>-Hypergeometric Polynomials and Associated Generating Functions |
| title_short | A General Family of <i>q</i>-Hypergeometric Polynomials and Associated Generating Functions |
| title_sort | general family of i q i hypergeometric polynomials and associated generating functions |
| topic | basic (or <i>q</i>-) hypergeometric series homogeneous <i>q</i>-difference operator <i>q</i>-binomial theorem cauchy polynomials Al-Salam-Carlitz <i>q</i>-polynomials rogers type formulas |
| url | https://www.mdpi.com/2227-7390/9/11/1161 |
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