Generalized <i>q</i>-Difference Equations for <i>q</i>-Hypergeometric Polynomials with Double <i>q</i>-Binomial Coefficients
In this paper, we apply a general family of basic (or <i>q</i>-) polynomials with double <i>q</i>-binomial coefficients as well as some homogeneous <i>q</i>-operators in order to construct several <i>q</i>-difference equations involving seven variables...
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MDPI AG
2022-02-01
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| Online Access: | https://www.mdpi.com/2227-7390/10/4/556 |
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| author | Jian Cao Hari M. Srivastava Hong-Li Zhou Sama Arjika |
| author_facet | Jian Cao Hari M. Srivastava Hong-Li Zhou Sama Arjika |
| author_sort | Jian Cao |
| collection | DOAJ |
| description | In this paper, we apply a general family of basic (or <i>q</i>-) polynomials with double <i>q</i>-binomial coefficients as well as some homogeneous <i>q</i>-operators in order to construct several <i>q</i>-difference equations involving seven variables. We derive the Rogers type and the extended Rogers type formulas as well as the Srivastava-Agarwal-type bilinear generating functions for the general <i>q</i>-polynomials, which generalize the generating functions for the Cigler polynomials. We also derive a class of mixed generating functions by means of the aforementioned <i>q</i>-difference equations. The various results, which we have derived in this paper, are new and sufficiently general in character. Moreover, the generating functions presented here are potentially applicable not only in the study of the general <i>q</i>-polynomials, which they have generated, but indeed also in finding solutions of the associated <i>q</i>-difference equations. Finally, we remark that it will be a rather trivial and inconsequential exercise to produce 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 forced-in parameter <i>p</i> is obviously redundant. |
| first_indexed | 2024-03-09T21:31:00Z |
| format | Article |
| id | doaj.art-f44fde131ac8454a88aaaa4b67d2e542 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T21:31:00Z |
| publishDate | 2022-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-f44fde131ac8454a88aaaa4b67d2e5422023-11-23T20:56:33ZengMDPI AGMathematics2227-73902022-02-0110455610.3390/math10040556Generalized <i>q</i>-Difference Equations for <i>q</i>-Hypergeometric Polynomials with Double <i>q</i>-Binomial CoefficientsJian Cao0Hari M. Srivastava1Hong-Li Zhou2Sama Arjika3School of Mathematics, Hangzhou Normal University, Hangzhou 311121, ChinaDepartment of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, CanadaSchool of Mathematics, Hangzhou Normal University, Hangzhou 311121, ChinaDepartment of Mathematics and Informatics, University of Agadez, Agadez P.O. Box 199, NigerIn this paper, we apply a general family of basic (or <i>q</i>-) polynomials with double <i>q</i>-binomial coefficients as well as some homogeneous <i>q</i>-operators in order to construct several <i>q</i>-difference equations involving seven variables. We derive the Rogers type and the extended Rogers type formulas as well as the Srivastava-Agarwal-type bilinear generating functions for the general <i>q</i>-polynomials, which generalize the generating functions for the Cigler polynomials. We also derive a class of mixed generating functions by means of the aforementioned <i>q</i>-difference equations. The various results, which we have derived in this paper, are new and sufficiently general in character. Moreover, the generating functions presented here are potentially applicable not only in the study of the general <i>q</i>-polynomials, which they have generated, but indeed also in finding solutions of the associated <i>q</i>-difference equations. Finally, we remark that it will be a rather trivial and inconsequential exercise to produce 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 forced-in parameter <i>p</i> is obviously redundant.https://www.mdpi.com/2227-7390/10/4/556homogeneous <i>q</i>-difference operatordouble <i>q</i>-binomial coefficients<i>q</i>-difference equations<i>q</i>-hypergeometric polynomialsgenerating functions |
| spellingShingle | Jian Cao Hari M. Srivastava Hong-Li Zhou Sama Arjika Generalized <i>q</i>-Difference Equations for <i>q</i>-Hypergeometric Polynomials with Double <i>q</i>-Binomial Coefficients Mathematics homogeneous <i>q</i>-difference operator double <i>q</i>-binomial coefficients <i>q</i>-difference equations <i>q</i>-hypergeometric polynomials generating functions |
| title | Generalized <i>q</i>-Difference Equations for <i>q</i>-Hypergeometric Polynomials with Double <i>q</i>-Binomial Coefficients |
| title_full | Generalized <i>q</i>-Difference Equations for <i>q</i>-Hypergeometric Polynomials with Double <i>q</i>-Binomial Coefficients |
| title_fullStr | Generalized <i>q</i>-Difference Equations for <i>q</i>-Hypergeometric Polynomials with Double <i>q</i>-Binomial Coefficients |
| title_full_unstemmed | Generalized <i>q</i>-Difference Equations for <i>q</i>-Hypergeometric Polynomials with Double <i>q</i>-Binomial Coefficients |
| title_short | Generalized <i>q</i>-Difference Equations for <i>q</i>-Hypergeometric Polynomials with Double <i>q</i>-Binomial Coefficients |
| title_sort | generalized i q i difference equations for i q i hypergeometric polynomials with double i q i binomial coefficients |
| topic | homogeneous <i>q</i>-difference operator double <i>q</i>-binomial coefficients <i>q</i>-difference equations <i>q</i>-hypergeometric polynomials generating functions |
| url | https://www.mdpi.com/2227-7390/10/4/556 |
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