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analytic continuation analytic function Arthur asymptotic expansion Bailey Bateman behavior Bernoulli Bernoulli polynomials Bessel functions Burchnall Calcutta Math Chapter Chaundy coefficients confluent hypergeometric functions converges cosh defined denotes derived differential equation Erdelyi Euler's finite follows func G function G. N. Watson gamma function Gauss Goursat Hence hyper hypergeo hypergeometric equation hypergeometric series integer integral representations integrand investigated Laguerre polynomials large number Legendre functions linear linearly independent linearly independent solutions London Math loop lower sign MacRobert mathematical Meijer notation obtain one-valued Oxford parameters path of integration Philos Pochhammer poles polynomials positive integer power series Proc proved quadratic transformations Quart Read real axis recurrence relations residues respectively Riemann's right-hand side satisfies Shabde singularities sinh substitution term by term theorem theory tion Tricomi upper or lower valid values variable Whittaker zero