In this paper, we derive an integral transform involving the product of Hermite polynomial $ H_{n}(\beta x) $ and parabolic cylinder function $ D_{v}(\alpha t) $. These integral transforms will be evaluated in terms of Lerch function. Various formulae are also evaluated in terms of special functions to complete this paper. All the results in this paper are new.
Citation: Robert Reynolds, Allan Stauffer. A quintuple integral involving the product of Hermite polynomial $ H_{n}(\beta x) $ and parabolic cylinder function $ D_{v}(\alpha t) $: derivation and evaluation[J]. AIMS Mathematics, 2022, 7(5): 7464-7470. doi: 10.3934/math.2022418
In this paper, we derive an integral transform involving the product of Hermite polynomial $ H_{n}(\beta x) $ and parabolic cylinder function $ D_{v}(\alpha t) $. These integral transforms will be evaluated in terms of Lerch function. Various formulae are also evaluated in terms of special functions to complete this paper. All the results in this paper are new.
| [1] |
S. Goldstein, Operational representations of Whittaker's confluent hypergeometric function and Weber's parabolic cylinder function, Proc. London math. Soc., s2-34 (1932), 103–125. http://dx.doi.org/10.1112/plms/s2-34.1.103 doi: 10.1112/plms/s2-34.1.103
|
| [2] | R. S. Varma, An integral equation for the Weber-Hermite functions, Tohoku Math. J., 35 (1932), 323–325. |
| [3] |
S. Mitra, On the squares of Weber's parabolic cylinder functions and certain integrals connected with them, Proc. Edinburgh Math. Soc., 4 (1934), 27–32. http://dx.doi.org/10.1017/S0013091500024147 doi: 10.1017/S0013091500024147
|
| [4] |
D. J. Daniel, Orthogonal representation of Weber's function using Hermite polynomials, J. Approx. Theory, 113 (2001), 156–163. http://dx.doi.org/10.1006/jath.2001.3620 doi: 10.1006/jath.2001.3620
|
| [5] |
G. A. Martynov, G. N. Sarkisov, Exact equations and the theory of liquids. V, Mol. Phys., 49 (1983), 1495–1504. http://dx.doi.org/10.1080/00268978300102111 doi: 10.1080/00268978300102111
|
| [6] |
R. Reynolds, A. Stauffer, A method for evaluating definite integrals in terms of special functions with examples, Int. Math. Forum, 15 (2020), 235–244. http://dx.doi.org/10.12988/imf.2020.91272 doi: 10.12988/imf.2020.91272
|
| [7] | Yu. A. Brychkov, O. I. Marichev, N. V. Savischenko, Handbook of Mellin transforms, CRC Press, 2019. |
| [8] | I. S. Gradshteyn, I. M. Ryzhik, Tables of integrals, series and products, 6 Eds., Cambridge, MA: Academic Press, 2000. |
| [9] | H. M. Srivastava, T. Kim, Y. Simsek, q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic $L$-series, Russ. J. Math. Phys., 12 (2005), 241–268. |
| [10] | National Institute of Standards and Technology, NIST Digital Library of Mathematical Functions, 2010. Available from: https://dlmf.nist.gov/. |
| [11] | K. B. Oldham, J. C. Myland, J. Spanier, An atlas of functions: with equator, the atlas function calculator, 2 Eds., New York, NY: Springer, 2009. http://dx.doi.org/10.1007/978-0-387-48807-3 |
Robert Reynolds, Allan Stauffer. A quintuple integral involving the product of Hermite polynomial $ H_{n}(\beta x) $ and parabolic cylinder function $ D_{v}(\alpha t) $: derivation and evaluation[J]. AIMS Mathematics, 2022, 7(5): 7464-7470. doi: 10.3934/math.2022418
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