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References
M. Abramowitz and I.A. Stegun, editors, Handbook of mathematical functions, Dover, New York, 1965.
G.E. Andrews, An incredible formula of Ramanujan, Australian Math. Soc. Gazette 6 (1979), 80–89.
P. Appell, Sur une classe de fonctions analogues aux fonctions Eulériennes, Math. Ann. 19 (1882), 84–102.
R. Ayoub, An introduction to the analytic theory of numbers, American Mathematical Society, Providence, 1963.
R. Ayoub, Euler and the zeta function, Amer. Math. Monthly 81 (1974), 1067–1086.
B.C. Berndt, Elementary evaluation of ξ(2n), Math. Mag. 48 (1975), 148–154.
B.C. Berndt, P.T. Joshi, and B.M. Wilson, Chapter 2 of Ramanujan's second notebook, Glasgow Math. J. (to appear).
B.C. Berndt and L. Schoenfeld, Periodic analogues of the Euler-Maclaurin and Poisson summation formulas with applications to number theory, Acta Arith. 28 (1975), 23–68.
T.B.N. Brodén, Bemerkungen über sogenannte finite Integration, Arkiv för Mat. Astr. och Fys. (Stockholm) 7 (1911), 34pp.
T.B.N. Brodén, Einige Anwendungen diskontinuierlicher Integrale auf Fragen der Differenzenrechnung, Acta Univ. Lund. (2) 8 (1912), 17pp.
T.J. I'A. Bromwich, An introduction to the theory of infinite series, second ed., Macmillan, London, 1926.
L. Carlitz, Eulerian numbers and polynomials, Math. Mag. 33 (1959), 247–260.
L. Carlitz, Eulerian numbers and operators, Coll. Math. 24 (1973), 175–200.
L. Carlitz, Some remarks on the Eulerian function, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 611, 1978, 79–91.
L. Carlitz, D.C. Kurtz, R. Scoville, and O.P. Stackelberg, Asymptotic properties of Eulerian numbers, Z. Wahrscheinlichkeitstheorie 23 (1972), 47–54.
L. Carlitz and J. Riordan, Congruences for Eulerian numbers, Duke Math. J. 20 (1953), 339–343.
L. Carlitz and R. Scoville, Generalized Eulerian numbers: combinatorial applications, J. Reine Angew. Math. 265 (1974), 110–137 Corrigendum 288 (1976), 218–219.
L. Carlitz and R. Scoville, Eulerian numbers and operators, Fibonacci Quart. 13 (1975), 71–83.
L. Euler, Institutiones calculi differentialis, Acad. Imperialis Sci., Petrograd, 1755.
F.G. Frobenius, Über die Bernoullischen Zahlen und die Eulerschen Polynome, Sitz. d. K. Preuss. Akad. Wiss. Berlin 1910, 809–847.
F.G. Frobenius, Gesammelte Abhandlungen, Band III, Springer-Verlag, Berlin, 1968.
E. Landau, Primzahlen, Chelsea, New York, 1953.
A.M. Legendre, Traité des fonctions elliptiques et des intégrales Eulériennes, tome II, Huzard-Courcier, Paris, 1826.
N. Nielsen, Traité élémentaire des nombres de Bernoulli, Gauthier-Villars, Paris, 1923.
N.E. Nörlund, IIC7. Neure Untersuchungen über Differenzengleichungen, Encyklopädie der mathematischen Wissenschaften, Band 2, Teil 3, B.G. Teubner, Leipzig, 1923, pp. 675–721.
N.E. Nörlund, Vorlesungen über Differenzenrechnung, Chelsea, New York, 1954.
E. Picard, Sur une classe de transcendantes nouvelles, Acta Math. 18 (1894), 133–154.
S. Ramanujan, Some properties of Bernoulli's numbers, J. Indian Math. Soc. 3 (1911), 219–234.
S. Ramanujan, Irregular numbers, J. Indian Math. Soc. 5 (1913), 105–106.
S. Ramanujan, Collected papers, Chelsea, New York, 1962.
S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957.
J. Riordan, An introduction to combinatorial analysis, John Wiley, New York, 1958.
J. Riordan, Combinatorial identities, John Wiley, New York, 1968.
T. J. Stieltjes, Table des valeurs des sommes \(S_k = \sum\limits_1^\infty {n^{ - k} }\), Acta Math. 10 (1887), 299–302.
J. V. Uspensky and M. A. Heaslet, Elementary number theory, McGraw-Hill, New York, 1939.
S. S. Wagstaff, Jr., The irregular primes to 125000, Math. Comp. 32 (1978), 583–591.
S. S. Wagstaff, Jr., Ramanujan's paper on Bernoulli numbers, submitted for publication.
J. Worpitzky, Studien über die Bernoullischen und Eulerschen Zahlen, J. Reine Angew. Math. 94 (1883), 203–232.
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Dedicated to Emil Grosswald, with respect and admiration.
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Berndt, B.C., Wilson, B.M. (1981). Chapter 5 Ramanujan's second notebook. In: Knopp, M.I. (eds) Analytic Number Theory. Lecture Notes in Mathematics, vol 899. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0096453
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