Solution of Some Integral Equations Involving Confluent *k*-Hypergeometric Functions

Author(s)
Shahid Mubeen

ABSTRACT

The principle aim of this research article is to investigate the properties of k-fractional integration introduced and defined by Mubeen and Habibullah [1],and secondly to solve the integral equation of the form

, for k > 0, β > 0, y > 0, 0 < x < t < ∞, where is the confluent k-hypergeometric functions, by using k-fractional integration.

The principle aim of this research article is to investigate the properties of k-fractional integration introduced and defined by Mubeen and Habibullah [1],and secondly to solve the integral equation of the form

, for k > 0, β > 0, y > 0, 0 < x < t < ∞, where is the confluent k-hypergeometric functions, by using k-fractional integration.

Cite this paper

S. Mubeen, "Solution of Some Integral Equations Involving Confluent*k*-Hypergeometric Functions," *Applied Mathematics*, Vol. 4 No. 7, 2013, pp. 9-11. doi: 10.4236/am.2013.47A003.

S. Mubeen, "Solution of Some Integral Equations Involving Confluent

References

[1] S. Mubeen and G. M. Habibullah, “k-Fractional Integrals and Application,” International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 2, 2012, pp. 89-94.

[2] A. Erdélyi, “An Integral Equation Involving Legendre Functions,” Journal of the Society for Industrial and Applied Mathematics, Vol. 12, No. 1, 1964, pp. 15-30. doi:10.1137/0112002

[3] E. R. Love, “Some Integral Equations Involving Hypergeometric Functions,” Proceedings of the Edinburgh Mathematical Society, Vol. 15, No. 3, 1967, pp. 169-198. doi:10.1017/S0013091500011706

[4] G. M. Habibullah, “Some Integral Equations Involving Confluent Hypergeometric Functions,” The Yokohama Mathematical Journal, Vol. 19, 1971, pp. 35-43.

[5] K. N. Srivastava, “Fractional Integration and Integral Equations with Polynomial Kernels,” Journal of the Society for Industrial and Applied Mathematics, Vol. 40, 1965, pp. 435-440.

[6] R. Diaz and C. Teruel, “q,k-Generalized Gamma and Beta Functions,” Journal of Nonlinear Mathematical Physics, Vol. 12, No. 1, 2005, pp. 118-134. doi:10.2991/jnmp.2005.12.1.10

[7] R. Diaz and E. Pariguan, “On Hypergeometric Functions and Pochhammer k-Symbol,” Divulgaciones Matemáticas, Vol. 15, No. 2, 2007, pp. 179-192.

[8] R. Diaz, C. Ortiz and E. Pariguan, “On the k-Gamma q-Distribution,” Central European Journal of Mathematics, Vol. 8, No. 3, 2010, pp. 448-458. doi:10.2478/s11533-010-0029-0

[9] M. Mansour, “Determining the k-Generalized Gamma Function Γk(x) by Functional Equations,” International Journal of Contemporary Mathematical Sciences, Vol. 4, No. 21, 2009, pp. 1037-1042.

[10] 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, 2010, pp. 653-660.

[11] V. Krasniqi, “A Limit for the k-Gamma and k-Beta Function,” International Mathematical Forum, Vol. 5, No. 33, 2010, pp. 1613-1617.

[12] V. Krasniqi, “Inequalities and Monotonicity for the Ration of k-Gamma Function,” Scientia Magna, Vol. 6, No. 1, 2010, pp. 40-45.

[13] F. Merovci, “Power Product Inequalities for the Γk Function,” International Journal of Mathematical Analysis, Vol. 4, No. 21-24, 2010, pp. 1007-1012.

[14] S. Mubeen and G. M. Habibullah, “An Integral Representation of Some k-Hypergeometric Functions,” International Mathematical Forum, Vol. 7, No. 4, 2012, pp. 203-207.

[15] S. Mubeen, “k-Analogue of Kummer’s First Formula,” Journal of Inequalities and Special Functions, Vol. 3, No. 3, 2012, pp. 41-44.

[1] S. Mubeen and G. M. Habibullah, “k-Fractional Integrals and Application,” International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 2, 2012, pp. 89-94.

[2] A. Erdélyi, “An Integral Equation Involving Legendre Functions,” Journal of the Society for Industrial and Applied Mathematics, Vol. 12, No. 1, 1964, pp. 15-30. doi:10.1137/0112002

[3] E. R. Love, “Some Integral Equations Involving Hypergeometric Functions,” Proceedings of the Edinburgh Mathematical Society, Vol. 15, No. 3, 1967, pp. 169-198. doi:10.1017/S0013091500011706

[4] G. M. Habibullah, “Some Integral Equations Involving Confluent Hypergeometric Functions,” The Yokohama Mathematical Journal, Vol. 19, 1971, pp. 35-43.

[5] K. N. Srivastava, “Fractional Integration and Integral Equations with Polynomial Kernels,” Journal of the Society for Industrial and Applied Mathematics, Vol. 40, 1965, pp. 435-440.

[6] R. Diaz and C. Teruel, “q,k-Generalized Gamma and Beta Functions,” Journal of Nonlinear Mathematical Physics, Vol. 12, No. 1, 2005, pp. 118-134. doi:10.2991/jnmp.2005.12.1.10

[7] R. Diaz and E. Pariguan, “On Hypergeometric Functions and Pochhammer k-Symbol,” Divulgaciones Matemáticas, Vol. 15, No. 2, 2007, pp. 179-192.

[8] R. Diaz, C. Ortiz and E. Pariguan, “On the k-Gamma q-Distribution,” Central European Journal of Mathematics, Vol. 8, No. 3, 2010, pp. 448-458. doi:10.2478/s11533-010-0029-0

[9] M. Mansour, “Determining the k-Generalized Gamma Function Γk(x) by Functional Equations,” International Journal of Contemporary Mathematical Sciences, Vol. 4, No. 21, 2009, pp. 1037-1042.

[10] 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, 2010, pp. 653-660.

[11] V. Krasniqi, “A Limit for the k-Gamma and k-Beta Function,” International Mathematical Forum, Vol. 5, No. 33, 2010, pp. 1613-1617.

[12] V. Krasniqi, “Inequalities and Monotonicity for the Ration of k-Gamma Function,” Scientia Magna, Vol. 6, No. 1, 2010, pp. 40-45.

[13] F. Merovci, “Power Product Inequalities for the Γk Function,” International Journal of Mathematical Analysis, Vol. 4, No. 21-24, 2010, pp. 1007-1012.

[14] S. Mubeen and G. M. Habibullah, “An Integral Representation of Some k-Hypergeometric Functions,” International Mathematical Forum, Vol. 7, No. 4, 2012, pp. 203-207.

[15] S. Mubeen, “k-Analogue of Kummer’s First Formula,” Journal of Inequalities and Special Functions, Vol. 3, No. 3, 2012, pp. 41-44.