An alternative integral representation for the product of two parabolic cylinder functions. Issue 12 (2nd December 2017)

• Record Type: Journal Article
• Authors: Veestraeten, Dirk
• Is Part Of: Integral transforms and special functions. Volume 28:Issue 12(2017)
• Page Start: 915
• Page End: 922
• Publication Date: 2017-12-02
• Subjects: Error function -- integral representation -- Laplace transform -- modified Bessel function -- Nicholson-type integral -- parabolic cylinder function; 30E20 -- 33B20 -- 33C15 -- 44A10 -- 44A20; Integral transforms -- Periodicals; Transcendental functions -- Periodicals; Transformations (Mathematics) -- Periodicals; Calculus, Integral -- Periodicals; 515
• DOI: 10.1080/10652469.2017.1385612
• Languages: English
• ISSNs: 1065-2469

Abstract

ABSTRACT: Recently, [Veestraeten D. An integral representation for the product of parabolic cylinder functions. Integral Transforms Spec Funct. 2017;28(1):15–21] derived an integral representation forD ν ( x ) D μ ( y ) withRe ⁡ ( ν + μ ) < 1 that was expressed in terms of the Gaussian hypergeometric function. This paper obtains an alternative expression forD ν ( x ) D μ ( y ) in which the integrand contains the parabolic cylinder function itself with the condition for convergence being atRe ⁡ ( ν ) < 0 . The latter property is subsequently used to generate a new integral representation forD μ ( y ) in which restrictions on the order μ are absent.