The Journal of the Indian Mathematical Society

Open Access Open Access  Restricted Access Subscription Access
Open Access Open Access Open Access  Restricted Access Restricted Access Subscription Access

A New Class of Functions Suggested by the Generalized Basic Hypergeometric Function


Affiliations
1 Department of Mathematics, The Maharaja Sayajirao University of Baroda,Vadodara-390 002, India
2 Department of Mathematics, The Maharaja Sayajirao University of Baroda, Vadodara-390 002, India
     

   Subscribe/Renew Journal


We introduce an extended generalized basic hypergeometric function rΦs+p in which p tends to infinity together with the summation index. We define the difference operators and obtain infinite order difference equation, for which these new special functions are eigen functions. We derive some properties, as the order zero of this function, differential equation involving a particular hyper-Bessel type operators of infinite order, and contiguous function relations. A transformation formula and an l-analogue of the q-Maclaurin's series are also obtained.

Keywords

Basic Hypergeometric Function, q-Derivative, q-Integral, Eigen Function, Infinite Order Difference Equation.
Subscription Login to verify subscription
User
Notifications
Font Size


  • M. H. Annaby and Z. S. Mansour. q-Fractional Calculus and Equations. New York, Springer, 2012.

  • R. P. Boas. Entire Functions. Academic Press, 1954.

  • Meera H. Chudasama and B. I. Dave. A new class of functions suggested by the qhypergeometric function. The Mathematics Student, 85(3-4), (2016), 47-61.

  • G. Gasper and M. Rahman. Basic hypergeometric Series. Cambridge University press, Cambridge, 1990.

  • F. H. Jackson. q-form of Taylor's theorem. Messenger Math., 39, (1990), 62-64.

  • V. Kiryakova. Generalized Fractional Calculus and Applications. Longman and J. Wiley, Harlow and N. York, 1994.

  • V. Kiryakova. Transmutation method for solving hyper-Bessel dierential equations based on the Poisson-Dimovski transformation. Fractional Calculas and Applied Analysis, 11(3), (2008), 299-316.

  • V. Kiryakova. From the hyper-Bessel operators of Dimovski to the generalized fractional calculus. Fractional Calculus and Applied Analysis, 17(4), (2014), 977-1000.

  • B. Ya. Levin. Lectures on Entire Functions. Amer. Math. Soc., 1996.

  • M. Mansour. An asymptotic expansion of the q-gamma function q(x). Journal of Non-linear Mathematical Physics, 13(4), (2006), 479-483.

  • P. C. Sikkema. Dierential Operators and Equations. P. Noordho N. V., Groningen-Djakarta, 1953.

  • Rene F. Swarttouw. The contiguous function relations for the basic hypergeometric series. J. Math. Anal. Appl., 149, (1990), 151-159.


Abstract Views: 288

PDF Views: 1




  • A New Class of Functions Suggested by the Generalized Basic Hypergeometric Function

Abstract Views: 288  |  PDF Views: 1

Authors

Meera H. Chudasama
Department of Mathematics, The Maharaja Sayajirao University of Baroda,Vadodara-390 002, India
B. I. Dave
Department of Mathematics, The Maharaja Sayajirao University of Baroda, Vadodara-390 002, India

Abstract


We introduce an extended generalized basic hypergeometric function rΦs+p in which p tends to infinity together with the summation index. We define the difference operators and obtain infinite order difference equation, for which these new special functions are eigen functions. We derive some properties, as the order zero of this function, differential equation involving a particular hyper-Bessel type operators of infinite order, and contiguous function relations. A transformation formula and an l-analogue of the q-Maclaurin's series are also obtained.

Keywords


Basic Hypergeometric Function, q-Derivative, q-Integral, Eigen Function, Infinite Order Difference Equation.

References