Asymptotic expansions and analytic continuations for a class of Barnes-integrals

L.V. Ahlfors [1] Complex Analysis. New York, 1953. | Zbl 0052.07002

E.W. Barnes [2] The asymptotic expansion of integral functions defined by Taylor's series. Phil. Trans. Roy. Soc. London (A), 206, 249-297 (1906). | JFM 37.0427.01

[3] The asymptotic expansion of integral functions defined by generalized hypergeometric series. Proc. London Math. Soc. (2), 5, 59-116 (1907). | JFM 38.0449.01

[4] A new development of the theory of the hypergeometric functions. Proc. London Math. Soc. (2), 6, 141-177 (1908). | JFM 39.0506.01

S. Bochner [5] Some properties of modular relations. Ann. of Math. 53, 332-363 (1951). | Zbl 0042.32101

[5a] Theorems on analytic continuation which occur in the study of Riemann's functional equation. Journ. Indian Math. Soc., New Series 21, 127-147 (1957). | Zbl 0096.05203

[6] On Riemann's functional equation with multiple gamma factors. Ann. of Math. 67, 29-41 (1958). | Zbl 0082.29002

J. Boersma [7] On a function, which is a special case of Meijer's G-function. Compos. Math. 15, 34-63 (1962). | Numdam | Zbl 0100.06505

T.J. I' A. Bromwich [8] An introduction to the theory of infinite series. 2nd ed., London, 1949. | JFM 39.0306.02 | Zbl 0004.00705

K. Chandrasekharan and Raghavan Narasimhan [9] Functional equations with multiple gamma factors and the average order of arithmetical functions. Ann. of Math. 76, 93-136 (1962). | Zbl 0211.37901

E.T. Copson [10] An introduction to the theory of functions of a complex variable. Oxford, 1950. | Zbl 0012.16902

J.G. Van Der Corput [11] On the coefficients in certain asymptotic factorial expansions I, II. Proc. Kon. Ned. Akad. Wet. Ser. A, 60, 337-351 (1957). | Zbl 0079.09903

A.L. Dixon and W.L. Ferrar [12] A class of discontinuous integrals. Quart. Journ. Math. Oxford Ser. 7, 81-96 (1936). | JFM 62.0480.01 | Zbl 0014.10804

G. Doetsch [13] Handbuch der Laplace-Transformation I, II Basel, 1950, 1955. | Zbl 0040.05901

A. Erdélyi [14] Asymptotic expansions. New York, 1956. | MR 78494 | Zbl 0070.29002

A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi [15] Higher transcendental functions I. New York, 1953. | Zbl 0051.30303

W.B. Ford [16] The asymptotic developments of functions defined by Maclaurin series. Ann Arbor, 1936. | JFM 62.1203.03 | Zbl 0016.12402

C. Fox [17] The asymptotic expansion of generalized hypergeometric functions. Proc. London Math. Soc. (2), 27, 389-400 (1928). | JFM 54.0392.03

[18] The G and H functions as symmetrical Fourier kernels. Trans. Amer. Math. Soc. 98, 395-429 (1961). | Zbl 0096.30804

H.K. Hughes [19] On the asymptotic expansions of entire functions defined by Maclaurin series. Bull. Amer. Math. Soc. 50, 425-430 (1944). | MR 9987 | Zbl 0063.02937

[20] The asymptotic developments of a class of entire functions. Bull. Amer. Math. Soc. 51, 456-461 (1945). | Zbl 0063.02938

T.M. Macrobert [21 ] Induction proofs of the relations between certain asymptotic expansions and corresponding generalized hypergeometric series. Proc. Roy. Soc. Edinburgh 58, 1-13 (1938). | JFM 64.0337.01 | Zbl 0018.06101

C.S. Meijer [22] On the G-function I-VIII. Proc. Kon. Ned. Akad. Wet. 49, 227-237, 344-356, 457-469, 632-641, 765-772, 936-943, 1063-1072, 1165-1175 (1946). | Zbl 0060.19901

C.V. Newsom [23] On the character of certain entire functions in distant portions of the plane. Amer. J. Math. 60, 561-572 (1938). | JFM 64.0292.04 | Zbl 0019.17106

[24] The asymptotic behavior of a class of entire functions. Amer. J. Math. 65, 450-454 (1943). | Zbl 0063.05942

F.W.J. Olver [25] Note on the asymptotic expansion of generalized hypergeometric functions. Journ. London Math. Soc. 28, 462-464 (1953). | MR 56749 | Zbl 0052.06903

T.D. Riney [26] On the coefficients in asymptotic factorial expansions. Proc. Amer. Math. Soc. 7, 245-249 (1956). | MR 76903 | Zbl 0072.06804

C.H. Rowe [27] A proof of the asymptotic series for log Γ(z) and log Γ(z+a). Ann. of Math. Second Ser. 32, 10-16 (1931). | JFM 57.0422.01 | Zbl 0001.21301

G. Sansone and J.C.H. Gerretsen [28] Lectures on the theory of functions of a complex variable I. Groningen, 1960. | Zbl 0093.26803

G.N. Watson [29] A class of integral functions defined by Taylor's series. Trans. Cambr. Phil. Soc. 22, 15-37 (1913). | JFM 44.0476.03

E.T. Whittaker and G.N. Watson [30] A course of modern analysis. 4th ed. Cambridge, 1958. | JFM 45.0433.02

E.M. Wright [31] The asymptotic expansion of the generalized Bessel function. Proc. London Math. Soc. (2), 38, 257-270 (1935). | JFM 60.0306.02 | Zbl 0010.21103

[32] The asymptotic expansion of the generalized hypergeometric function. Journ. London Math. Soc. 10, 286-293 (1935). | JFM 61.0407.01 | Zbl 0013.02104

[33] The asymptotic expansion of integral functions defined by Taylor series. Phil. Trans. Roy. Soc. London (A) 238, 423-451 (1940). | JFM 66.0352.01 | MR 1296 | Zbl 0023.14002

[34] The asymptotic expansion of the generalized hypergeometric function. Proc. London Math. Soc. (2) 46, 389-408 (1940). | JFM 66.1243.01 | MR 3876 | Zbl 0025.40402

[35] The generalized Bessel function of order greater than one. Quart. Journ. Math. Oxford Ser. 11, 36-48 (1940). | JFM 66.0322.01 | MR 3875 | Zbl 0023.14101

[36] The asymptotic expansion of integral functions defined by Taylor series (second paper). Phil. Trans. Roy. Soc. London (A) 239, 217-222 (1946). | Zbl 0061.14506

[37] A recursion formula for the coefficients in an asymptotic expansion. Proc. Glasgow Math. Ass. 4, 38-41 (1959-1960). | Zbl 0086.06101

D. Wrinch [38] An asymptotic formula for the hypergeometric function 0Δ4 (z). Philos. Magazine (6) 41, 161-173 (1921). | JFM 48.1244.01

[39] A generalized hypergeometric function with n parameters. Philos. Magazine (6) 41, 174-186 (1921).

[40] Some approximations to hypergeometric functions. Philos. Magazine (6) 45, 818-827 (1923). | JFM 49.0252.01