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Moment integrals of powers of Airy functions. (English) Zbl 0784.33003

An analytic approach to compute integrals: J n (α)= 0 z n A i ( z ) α dz, J n ' (α)= 0 A i ' (z) α dz is presented, where Ai ' [z] is the derivative of the Airy function Ai(z) and α is any real number. Its mathematical basis lies on the introduction of an auxiliary function:

i k,β (α)= 0 z k A i ( z ) β A i ' (z) α-β dz,withβα·

and reduction to a linear partial difference equation with two variables and then derivation of recurrence relations for J n and J n ' .

MSC:
33C10Bessel and Airy functions, cylinder functions, 0 F 1
References:
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[5]L. W. Pearson,A Scheme for Automatic Computation of Fock-type Integrals, Institute of Electrical and Electronic Engineers, Transactions on Antennas and Propagation35, 1111-1118 (1987). · Zbl 0946.78517 · doi:10.1109/TAP.1987.1143985
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[8]M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions, Natl. Bur. of Standards Applied Math. Series55, Chap. 10 (1964).
[9]R. E. Mickens,Difference Equations, Van Nostrand Reinhold, New York 1987.
[10]ibid., ref. 8., Chap. 15.
[11]ibid., ref. 8., Chap. 25.
[12]ibid., ref. 8., Chap. 6.