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VKiryakova_fcaa113

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302 V. Kiryakovareduces for m = 2 to the classical cosine function cos(x) = cos 2 (x) as asolution of y ′′ (x) = −y(x).For the Bessel function, the following integral representationJ ν (x) =2(x/2)ν √ πΓ(ν +1∫2 ) π/20cos(x sin ϕ)(cos ϕ) 2ν dϕ, R ν > −1/2, (11)is known as the Poisson integral (formula), see e.g. [12], Vol. 2, §7.12.After a substitution sin ϕ := t, it gets the form of a fractional order (ν+1/2)operator of integration (Erdélyi-Kober operator of Riemann-Lioville type):J ν (x) =2(x/2)ν √ πΓ(ν +1∫2 ) 10(1 − t 2 ) ν− 1 2 cos(xt)dt. (12)In our studies (as [10], [15, Chs. 3, 4], [16]) on hyper-Bessel differentialequations (8) and their solutions, expressed in terms of the hyper-Besselfunctions j ν (m−1)1 ,...,ν m−1(x) of Delerue [2], as analogues of the normalized Besselfunction j ν (x) in (9), we have introduced generalizations of the Poisson integralformula (12) of j ν (m−1)1 ,...,ν m−1(x) involving the generalized cosine function(10) of order m via operators of generalized fractional calculus, [15].2. Transmutation method and generalized Poissontransformation due to DimovskiThe essence of this method lies in the natural striving to find solutionsof new complicated problems by their reduction to well-known or simplerones, by means of a specific “translator”. In a narrow sense, the notionof transmutation operator originates from the works of Delsarte and Lions(1956 - 1959), see for example [4] and the posthumously published works[3], Vol. 1, p. 427. If ˜B : X −→ X and B : X −→ X are two operatorsacting in a space X , then the isomorphism T : X −→ X is said to transmute˜B into B, if the similarity relation holds: T ˜B = BT in X . In this sense,the transmutation method has been widely used in mathematical analysis,and mainly in solving differential equations and problems of mathematicalphysics (see Delsarte [3], Delsarte and Lions [4], Hearsh [13]. For applicationsin operational calculus, extended to so-called convolutional calculi, seefor example Dimovski [6], [7], [8]. Some of the authors use “similarities” (or”transmutations”) in a wider sense, as below.

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