Journal of Applied Mathematics

Solutions of k -Hypergeometric Differential Equations

Shahid Mubeen, Mammona Naz, Abdur Rehman, and Gauhar Rahman

Full-text: Open access

Abstract

We solve the second-order linear differential equation called the k -hypergeometric differential equation by using Frobenius method around all its regular singularities. At each singularity, we find 8 solutions corresponding to the different cases for parameters and modified our solutions accordingly.

Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 128787, 13 pages.

Dates
First available in Project Euclid: 2 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.jam/1425305724

Digital Object Identifier
doi:10.1155/2014/128787

Mathematical Reviews number (MathSciNet)
MR3206870

Zentralblatt MATH identifier
1318.33003

Citation

Mubeen, Shahid; Naz, Mammona; Rehman, Abdur; Rahman, Gauhar. Solutions of $k$ -Hypergeometric Differential Equations. J. Appl. Math. 2014 (2014), Article ID 128787, 13 pages. doi:10.1155/2014/128787. https://projecteuclid.org/euclid.jam/1425305724


Export citation

References

  • L. Euler, Institutiones Calculi Integralis, vol. 1 of Opera Omnia Series, 1769.
  • C. F. Gauss, Disquisitiones Generales Circa Seriem Infinitam, vol. 3, Werke, 1813.
  • E. E. Kummer, “Über die hypergeometrische Reihe,” Journal: Für die Reine und Angewandte Mathematik, vol. 15, pp. 39–83, 1836.
  • B. Dwork, “On Kummer's twenty-four solutions of the hypergeometric differential equation,” Transactions of the American Mathematical Society, vol. 285, no. 2, pp. 497–521, 1984.
  • R. Díaz and C. Teruel, “$q,k$-generalized gamma and beta functions,” Journal of Nonlinear Mathematical Physics, vol. 12, no. 1, pp. 118–134, 2005.
  • R. Díaz and E. Pariguan, “On hypergeometric functions and Pochhammer $k$-symbol,” Revista Matemática de la Universidad del Zulia: Divulgaciones Matemáticas, vol. 15, no. 2, pp. 179–192, 2007.
  • R. Díaz, C. Ortiz, and E. Pariguan, “On the $k$-gamma $q$-distribution,” Central European Journal of Mathematics, vol. 8, no. 3, pp. 448–458, 2010.
  • C. G. Kokologiannaki, “Properties and inequalities of generalized $k$-gamma, beta and zeta functions,” International Journal of Contemporary Mathematical Sciences, vol. 5, no. 13–16, pp. 653–660, 2010.
  • V. Krasniqi, “Inequalities and monotonicity for the ration of $k$-gamma functions,” Scientia Magna, vol. 6, no. 1, pp. 40–45, 2010.
  • V. Krasniqi, “A limit for the $k$-gamma and $k$-beta function,” International Mathematical Forum: Journal for Theory and Applications, vol. 5, no. 33–36, pp. 1613–1617, 2010.
  • S. Mubeen and G. M. Habibullah, “$k$-fractional integrals and application,” International Journal of Contemporary Mathematical Sciences, vol. 7, no. 1–4, pp. 89–94, 2012.
  • S. Mubeen and G. M. Habibullah, “An integral representation of some $k$-hypergeometric functions,” International Mathematical Forum: Journal for Theory and Applications, vol. 7, no. 1–4, pp. 203–207, 2012.
  • S. Mubeen, “Solution of some integral equations involving con uent $K$-hypergeometric functions,” Applied Mathematics, vol. 4, no. 7A, pp. 9–11, 2013.
  • I. N. Sneddon, Special Functions of Mathematical Physics and Chemistry, Oliver and Boyd, 1956. \endinput