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analytic continuation analytic function Arthur asymptotic expansion Bailey Bateman behavior Bernoulli Bernoulli polynomials Bessel functions Burchnall Calcutta Math Chapter CM CM coefficients confluent hypergeometric functions Confluent hypergeometric series converges cosh defined denotes derived Erdelyi Euler's Feriet finite follows func G function G. N. Watson gamma function Gauss Goursat Hence hypergeo hypergeometric equation hypergeometric series integer integral formulas integral representations integrand investigated Laguerre polynomials large number Legendre functions linear linearly independent linearly independent solutions log T(z London Math loop lower sign MacRobert mathematical notation obtain one-valued Oxford parameters path of integration Philos Pochhammer poles positive integer power series Proc proved quadratic transformations Quart real axis recurrence relations residues Riemann's right-hand side Shabde singularities sinh T. W. Chaundy theorem theory tions Tricomi upper or lower valid values variable Watson Whittaker zero