• Communications of the Korean Mathematical Society
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• v.36 no.4
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• pp.705-714
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• 2021

The Whittaker function and its diverse extensions have been actively investigated. Here we aim to introduce an extension of the Whittaker function by using the known extended confluent hypergeometric function 𝚽_p,v and investigate some of its formulas such as integral representations, a transformation formula, Mellin transform, and a differential formula. Some special cases of our results are also considered.

• Nonlinear Functional Analysis and Applications
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• v.29 no.1
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• pp.99-111
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• 2024

In this paper, a new seven-parameter Mittag-Leffler function of a single complex variable is proposed as a generalization of the standard Mittag-Leffler function, certain generalizations of Mittag-Leffler function, hypergeometric function and confluent hypergeometric function. Certain essential analytic properties are mainly discussed, such as radius of convergence, order, type, differentiation, Mellin-Barnes integral representation and Euler transform in the complex plane. Its relation to Fox-Wright function and H-function is also developed.

• Bulletin of the Korean Mathematical Society
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• v.38 no.3
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• pp.517-526
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• 2001

In this paper, we study the continuous wavelet transform on the Heisenberg group H$^n$, and describe the related continuous multiscale analysis. By using the wavelet packet transform we obtain a reconstruction formula on L$^2$(H$^n$).

• Kyungpook Mathematical Journal
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• v.19 no.1
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• pp.119-125
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• 1979

In the present paper we study a new integral transform whose kernel involves the product of two H-functions of two complex variables. Next, we establish an inversion formula for this new transform. On account of very general nature of its kernel, several other integral transforms studies earlier by many research workers viz., Bose (1952), Mukherji (1962), Nigam (1963), Rathie (1965), Singh (1969), Mittal & Goel (1973), and Gupta, Garg & Kalla (1975), follow as its particular cases.

• Communications of the Korean Mathematical Society
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• v.33 no.2
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• pp.619-630
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• 2018

In this paper, we obtain a (p, q)-extension of the Whittaker function $M_{k,{\mu}}(z)$ together with its integral representations, by using the extended confluent hypergeometric function of the first kind ${\Phi}_{p,q}(b;c;z)$ [recently extended by J. Choi]. Also, we give some of its main properties, namely the summation formula, a transformation formula, a Mellin transform, a differential formula and inequalities. In addition, our extension on Whittaker function finds interesting connection with the Laguerre polynomials.

• Journal of applied mathematics & informatics
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• v.28 no.1_2
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• pp.515-521
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• 2010

In multivariate analysis, the inversion formula of the Stieltjes transform is used to find the density of a spectral distribution of random matrices of sample covariance type. Let $B_n\;=\;\frac{1}{N}Y_nY_n^TT_n$ where $Y_n\;=\;[Y_{ij}]_{n\;{\times}\;N}$ is with independent, identically distributed entries and $T_n$ is an $n\;{\times}\;n$ symmetric non-negative definite random matrix independent of the $Y_{ij}$'s. In the present paper, using the inversion formula of the Stieltjes transform, we will find that the limiting distribution of $B_n$ has a continuous density function away from zero.

• Journal of the Korean Mathematical Society
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• v.41 no.2
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• pp.265-294
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• 2004

In this paper, using a simple formula, we evaluate the conditional Fourier-Feynman transforms and the conditional convolution products of cylinder type functions, and show that the conditional Fourier-Feynman transform of the conditional convolution product is expressed as a product of the conditional Fourier-Feynman transforms. Also, we evaluate the conditional Fourier-Feynman transforms of the functions of the forms exp {$\int_{O}^{T}$ $\theta$(s,$\chi$(s))ds}, exp{$\int_{O}^{T}$ $\theta$(s,$\chi$(s))ds}$\Phi$($\chi$(T)), exp{$\int_{O}^{T}$ $\theta$(s,$\chi$(s))d${\zeta}$(s)}, exp{$\int_{O}^{T}$ $\theta$(s,$\chi$(s))d${\zeta}$(s)}$\Phi$($\chi$(T)) which are of interest in Feynman integration theories and quantum mechanics.

• Bulletin of the Korean Mathematical Society
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• v.47 no.5
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• pp.1011-1023
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• 2010

In this paper we define the $L_p$-mixed curvature function of a convex body. We develop a formula connection the support function of $L_p$-mixed projection body with Fourier transform of the $L_p$-mixed curvature function. Using this formula we solve an analog of the Shephard projection problem for $L_p$-mixed projection bodies.

• East Asian mathematical journal
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• v.35 no.1
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• pp.23-30
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• 2019

Since Mittag-Leffler introduced the so-called Mittag-Leffler function, a number of its extensions have been investigated due mainly to their applications in a variety of research subjects. Shukla and Prajapati presented a lot of formulas involving a generalized Mittag-Leffler function in a systematic manner. Motivated mainly by Shukla and Prajapati's work, we aim to investigate a generalized multi-index Mittag-Leffler function and, among possible numerous formulas, choose to present several formulas involving this generalized multi-index Mittag-Leffler function such as a recurrence formula, derivative formula, three integral transformation formulas. The results presented here, being general, are pointed out to reduce to yield relatively simple formulas including known ones.

• Honam Mathematical Journal
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• v.41 no.1
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• pp.163-174
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• 2019

The veritable pursuit of this exegesis is to exhibit integrals affined with the Bessel-Struve kernel function, which are explicitly inscribed in terms of generalized (Wright) hypergeometric function and also the product of generalized (Wright) hypergeometric function with sum of two confluent hypergeometric functions. Somewhat integrals involving exponential functions, modified Bessel functions and Struve functions of order zero and one are also obtained as special cases of our chief results.