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The homogeneous q-difference operator

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Title: The homogeneous q-difference operator
Authors: Chen, William Y.C. chenstation@yahoo.com, Fu, Amy M.1 fmu@eyou.com, Zhang, Baoyin1 zby75@eyou.com
Source: Advances in Applied Mathematics. Nov2003, Vol. 31 Issue 4, p659. 10p.
Subjects: Differential operators, Binomial theorem, Polynomials, Operator equations
Abstract: We introduce a q-differential operator Dxy on functions in two variables which turns out to be suitable for dealing with the homogeneous form of the q-binomial theorem as studied by Andrews, Goldman, and Rota, Roman, Ihrig, and Ismail, et al. The homogeneous versions of the q-binomial theorem and the Cauchy identity are often useful for their specializations of the two parameters. Using this operator, we derive an equivalent form of the Goldman–Rota binomial identity and show that it is a homogeneous generalization of the q-Vandermonde identity. Moreover, the inverse identity of Goldman and Rota also follows from our unified identity. We also obtain the q-Leibniz formula for this operator. In the last section, we introduce the homogeneous Rogers–Szego¨ polynomials and derive their generating function by using the homogeneous q-shift operator. [Copyright &y& Elsevier]
Copyright of Advances in Applied Mathematics is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: The homogeneous <f>q</f>-difference operator
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  Data: <searchLink fieldCode="AR" term="%22Chen%2C+William+Y%2EC%2E%22">Chen, William Y.C.</searchLink><i> chenstation@yahoo.com</i><br /><searchLink fieldCode="AR" term="%22Fu%2C+Amy+M%2E%22">Fu, Amy M.</searchLink><relatesTo>1</relatesTo><i> fmu@eyou.com</i><br /><searchLink fieldCode="AR" term="%22Zhang%2C+Baoyin%22">Zhang, Baoyin</searchLink><relatesTo>1</relatesTo><i> zby75@eyou.com</i>
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  Data: <searchLink fieldCode="JN" term="%22Advances+in+Applied+Mathematics%22">Advances in Applied Mathematics</searchLink>. Nov2003, Vol. 31 Issue 4, p659. 10p.
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  Data: <searchLink fieldCode="DE" term="%22Differential+operators%22">Differential operators</searchLink><br /><searchLink fieldCode="DE" term="%22Binomial+theorem%22">Binomial theorem</searchLink><br /><searchLink fieldCode="DE" term="%22Polynomials%22">Polynomials</searchLink><br /><searchLink fieldCode="DE" term="%22Operator+equations%22">Operator equations</searchLink>
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  Data: We introduce a <f>q</f>-differential operator <f>Dxy</f> on functions in two variables which turns out to be suitable for dealing with the homogeneous form of the <f>q</f>-binomial theorem as studied by Andrews, Goldman, and Rota, Roman, Ihrig, and Ismail, et al. The homogeneous versions of the <f>q</f>-binomial theorem and the Cauchy identity are often useful for their specializations of the two parameters. Using this operator, we derive an equivalent form of the Goldman–Rota binomial identity and show that it is a homogeneous generalization of the <f>q</f>-Vandermonde identity. Moreover, the inverse identity of Goldman and Rota also follows from our unified identity. We also obtain the <f>q</f>-Leibniz formula for this operator. In the last section, we introduce the homogeneous Rogers–Szego¨ polynomials and derive their generating function by using the homogeneous <f>q</f>-shift operator. [Copyright &y& Elsevier]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Advances in Applied Mathematics is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1016/S0196-8858(03)00040-X
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      – Code: eng
        Text: English
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        PageCount: 10
        StartPage: 659
    Subjects:
      – SubjectFull: Differential operators
        Type: general
      – SubjectFull: Binomial theorem
        Type: general
      – SubjectFull: Polynomials
        Type: general
      – SubjectFull: Operator equations
        Type: general
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      – TitleFull: The homogeneous <f>q</f>-difference operator
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            NameFull: Chen, William Y.C.
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            NameFull: Fu, Amy M.
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              Text: Nov2003
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              Y: 2003
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