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

An analytic approach to compute integrals: \(J_ n(\alpha)=\int^ \infty_ 0z^ n\bigl[Ai(z)\bigr]^ \alpha dz\), \(J_ n'(\alpha)=\int_ 0^ \infty\bigl[Ai'(z)\bigr]^ \alpha dz\) is presented, where \(Ai'[z]\) is the derivative of the Airy function \(Ai(z)\) and \(\alpha\) is any real number. Its mathematical basis lies on the introduction of an auxiliary function: \[ i_{k,\beta}(\alpha)=\int^ \infty_ 0z^ k\bigl[Ai(z)\bigr]^ \beta\bigl[Ai'(z)\bigr] ^{\alpha- \beta}dz,\quad\text{with }\beta\leq\alpha. \] and reduction to a linear partial difference equation with two variables and then derivation of recurrence relations for \(J_ n\) and \(J_ n'\).

MSC:

33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
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References:

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[10] ibid., ref. 8., Chap. 15.
[11] ibid., ref. 8., Chap. 25.
[12] ibid., ref. 8., Chap. 6.
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