• Journal of the Chungcheong Mathematical Society
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• v.24 no.2
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• pp.187-204
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• 2011

B. C. Carlson [Some extensions of Lardner's relations between $_0F_3$ and Bessel functions, SIAM J. Math. Anal. 1(2) (1970), 232-242] presented several useful relations between Bessel and generalized hypergeometric functions that generalize some earlier results. Here, by simply splitting Srivastava's hypergeometric function $H_B$ into eight parts, we show how some useful and generalized relations between Srivastava's hypergeometric functions $H_B$ and $F^{(3)}$ can be obtained. These main results are shown to be specialized to yield certain relations between functions $_0F_1$, $_1F_1$, $_0F_3$, ${\Psi}_2$, and their products including different combinations with different values of parameters and signs of variables. We also consider some other interesting relations between the Humbert ${\Psi}_2$ function and $Kamp\acute{e}$ de $F\acute{e}riet$ function, and between the product of exponential and Bessel functions with $Kamp\acute{e}$ de $F\acute{e}riet$ functions.

• Kyungpook Mathematical Journal
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• v.18 no.2
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• pp.311-319
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• 1978

This paper deals with the evaluation of two definite integrals involving spheroidal wave function, H-function of two variables, and the generalized hypergeometric function. Also, an expansion formula for the product of generalized hypergeometric function and the H-function of two variables has been obtained. Since the H-function of two variables, spheroidal wave functions, and the generalized hypergeometric function may be transformed into a number of higher transcendental functions and polynomials, the results obtained in this paper include some known results as their particular cases. As an application of such results, a problem of heat conduction in a non-homogenous bar has been solved by using the generalized Legendre transform [9].

• Communications of the Korean Mathematical Society
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• v.33 no.4
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• pp.1285-1301
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• 2018

This paper is devoted to the study of $Tur{\acute{a}}n$-type inequalities for some well-known special functions such as Gauss hypergeometric functions, generalized complete elliptic integrals and confluent hypergeometric functions which are derived by using a new form of the Cauchy-Bunyakovsky-Schwarz inequality. We also apply these inequalities for some sample of interest such as incomplete beta function, incomplete gamma function, elliptic integrals and modified Bessel functions to obtain their corresponding $Tur{\acute{a}}n$-type inequalities.

• The Pure and Applied Mathematics
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• v.19 no.1
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• pp.43-57
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• 2012

By simply splitting the hypergeometric Saran function $F_E$ into eight parts, we show how some useful and generalized relations between $F_E$ and Srivas- tava's hypergeometric function $F^{(3)}$ can be obtained. These main results are shown to be specialized to yield certain relations between functions $_0F_1$, $_1F_1$, $_0F_3$, ${\Psi}_2$, and their products including different combinations with different values of parameters and signs of variables.

• Kyungpook Mathematical Journal
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• v.56 no.1
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• pp.131-136
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• 2016

The main object of this paper is to present two general integral formulas whose integrands are the integrand given in the integral formula (3) and a finite product of the generalized Bessel function of the first kind.

• Kyungpook Mathematical Journal
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• v.46 no.2
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• pp.235-245
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• 2006

Applications of Weyl fractional $q$-integral operator to various generalized basic hypergeometric functions including the basic analogue of Fox's H-function have been investigated in the present paper. Certain interesting special cases have also been deduced.

• Communications of the Korean Mathematical Society
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• v.25 no.2
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• pp.185-191
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• 2010

Srivastava noticed the existence of three additional complete triple hypergeometric functions $H_A$, $H_B$ and $H_C$ of the second order in the course of an extensive investigation of Lauricella's fourteen hypergeometric functions of three variables. In 2004, Rathie and Kim obtained four summation formulas containing a large number of very interesting reducible cases of Srivastava's triple hypergeometric series $H_A$ and $H_C$. Here we are also aiming at presenting two unified summation formulas (actually, including 62 ones) for some reducible cases of Srivastava's $H_C$ with the help of generalized Dixon's theorem and generalized Whipple's theorem on the sum of a $_3F_2$ obtained earlier by Lavoie et al.. Some special cases of our results are also considered.

• Communications of the Korean Mathematical Society
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• v.31 no.3
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• pp.591-601
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• 2016

Integral transforms and fractional integral formulas involving well-known special functions are interesting in themselves and play important roles in their diverse applications. A large number of integral transforms and fractional integral formulas have been established by many authors. In this paper, we aim at establishing some (presumably) new integral transforms and fractional integral formulas for the generalized hypergeometric type function which has recently been introduced by Luo et al. [9]. Some interesting special cases of our main results are also considered.

• Communications of the Korean Mathematical Society
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• v.31 no.2
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• pp.329-342
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• 2016

Motivated essentially by Brychkov's work [1], we evaluate some new integrals involving hypergeometric function and the logarithmic function (including those obtained by Brychkov[1], Choi and Rathie [3]), which are expressed explicitly in terms of Gamma, Psi and Hurwitz zeta functions suitable for numerical computations.

• Communications of the Korean Mathematical Society
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• v.28 no.2
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• pp.303-317
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• 2013

In present paper, we obtain functions $R_t(c,{\nu},a,b)$ and $R_t(c,-{\mu},a,b)$ by using generalized hypergeometric function. A recurrence relation, integral representation of the generalized hypergeometric function $_2R_1(a,b;c;{\tau};z)$ and some special cases have also been discussed.