[1] |
Abel, N. H., Démonstration d’une expression de laquelle la formule binome est un cas particulier, J. Reine Angew. Math., 1, 159-160 (1826) · ERAM 001.0019cj |
[2] |
Abel, N. H., (Oeuvres, Vol. 1 (1881), Grøndahl & Son: Grøndahl & Son Christiania) |
[3] |
(Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions (1965), Dover: Dover New York) · Zbl 0171.38503 |
[4] |
Andrews, G. E., The Theory of Partitions (1976), Addison-Wesley: Addison-Wesley Reading, Mass · Zbl 0371.10001 |
[5] |
Ayoub, R., An Introduction to the Analytic Theory of Numbers (1963), Amer. Math. Soc: Amer. Math. Soc Providence, RI · Zbl 0128.04303 |
[6] |
Balasubrahmanian, N., On the number defined by \(Nr = (1e) ∑^∞n = 0 (n^r\) n!)\), Math. Student, 18, 130-132 (1950) · Zbl 0045.01703 |
[7] |
Bell, E. T., Exponential polynomials, Ann. of Math., 35, 258-277 (1934) · Zbl 0009.21202 |
[8] |
Bell, E. T., The iterated exponential integers, Ann. of Math., 39, 539-557 (1938) · Zbl 0019.15001 |
[9] |
Berndt, B. C.; Joshi, P. T.; Wilson, B. M., Chapter 2 of Ramanujan’s second notebook, Glasgow Math. J., 22, 199-216 (1981) · Zbl 0463.40002 |
[10] |
Birkeland, R., Über die Auflösung algebraischer Gleichungen durch hypergeometrische Funktionen, Math. Z., 26, 566-578 (1927) · JFM 53.0332.03 |
[11] |
Bromwich, T. J.I’A, An Introduction to the Theory of Infinite Series (1926), Macmillan: Macmillan London · JFM 52.0500.03 |
[12] |
Browne, D. H., Problem E461, solution by H. W. Becker, Amer. Math. Monthly, 48, 701-703 (1941) |
[13] |
Carlitz, L., On some polynomials of Tricomi, Boll. Un. Mat. Ital., 13, 3, 58-64 (1958) · Zbl 0081.06601 |
[14] |
Carlitz, L., Some congruences involving sums of binomial coefficients, Duke Math. J., 27, 77-79 (1960) · Zbl 0090.25801 |
[15] |
Carlitz, L., Single variable Bell polynomials, Collect. Math., 4, 13-25 (1962) · Zbl 0109.02906 |
[16] |
Carlitz, L., Some remarks on the Bell numbers, Fibonacci Quart., 18, 66-73 (1980) · Zbl 0426.10013 |
[17] |
Cesàro, E., Sur une équation aux différences mèlées, Nouv. Ann. Math., 4, 3, 36-40 (1885) · JFM 17.0337.01 |
[18] |
Comtet, L., Advanced Combinatorics (1974), Reidel: Reidel Dordrecht · Zbl 0283.05001 |
[19] |
Dobinski, G., Summirung der Reihe \(∑ n^mn! für m = 1, 2, 3, 4, 5,…\), Arch. Math. Phys., 61, 333-336 (1877) · JFM 09.0178.04 |
[20] |
Duparc, J. A.; Lekkerkerker, C. G.; Peremans, W., An Elementary Proof of a Formula of Jensen, Math. Centrum Amsterdam, Rapport ZW 1952-021 (1952) · Zbl 0048.29703 |
[21] |
Eisenstein, G., Entwicklung von \(α^{α^α\) · ERAM 028.0815cj |
[22] |
Eisenstein, G., (Mathematische Werke, Vol. 1 (1975), Chelsea: Chelsea New York) · Zbl 0339.01018 |
[23] |
Epstein, L. F., A function related to the series for \(e^{e^x\) · Zbl 0021.41703 |
[24] |
Euler, L., De serie Lambertina plurimisque eius insignibus proprietatibus, (Opera Omnia, Bd. 6 (1921), B. G. Teubner: B. G. Teubner Leipzig), 350-369, Serie 1 |
[25] |
Feller, W., (An Introduction to Probability Theory and Its Applications, Vol. 2 (1966), Wiley: Wiley New York) · Zbl 0138.10207 |
[26] |
Ginsburg, J., Iterated exponentials, Scripta Math., 11, 340-353 (1945) · Zbl 0060.08415 |
[27] |
Gould, H. W., Some generalizations of Vandermonde’s convolution, Amer. Math. Monthly, 63, 84-91 (1956) · Zbl 0072.00702 |
[28] |
Gould, H. W., Final analysis of Vandermonde’s convolution, Amer. Math. Monthly, 64, 409-415 (1957) · Zbl 0079.01005 |
[29] |
Gould, H. W., Generalization of a theorem of Jensen concerning convolutions, Duke Math. J., 27, 71-76 (1960) · Zbl 0101.28702 |
[30] |
Gould, H. W., A series transformation for finding convolution identities, Duke Math. J., 28, 193-202 (1961) · Zbl 0122.30501 |
[31] |
Gould, H. W., A new convolution formula and some new orthogonal relations for the inversion of series, Duke Math. J., 29, 393-404 (1962) · Zbl 0122.30502 |
[32] |
Gould, H. W., Congruences involving sums of binomial coefficients and a formula of Jensen, Amer. Math. Monthly, 69, 400-402 (1962) · Zbl 0114.01105 |
[33] |
Gould, H. W., Brief guide to combinatorial addition theorems (1968), unpublished |
[34] |
Gould, H. W., Bell & Catalan Numbers (1976), Combinatorial Research Institute: Combinatorial Research Institute Morgantown |
[35] |
Hansen, E. R., A Table of Series and Products (1975), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0302.65039 |
[36] |
Hardy, G. H., On the zeroes of certain classes of integral Taylor series. II. On the integral function \(∑^∞n = 0 (x^n\)(n + a)\(^s\) n!)\) and other similar functions, (Proc. London Math. Soc. (2), 2 (1905)), 401-431 · JFM 36.0473.03 |
[37] |
Hardy, G. H., A Course of Pure Mathematics (1967), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0796.26002 |
[38] |
Hardy, G. H., Divergent Series (1949), Oxford Univ. Press (Clarendon): Oxford Univ. Press (Clarendon) London · Zbl 0032.05801 |
[39] |
Hardy, G. H., (Collected Papers, Vol. IV (1969), Oxford Univ. Press (Clarendon): Oxford Univ. Press (Clarendon) London) · Zbl 0181.28902 |
[40] |
Hardy, G. H., Ramanujan (1978), Chelsea: Chelsea New York · JFM 48.0019.04 |
[41] |
Jackson, F. H., A \(q\)-generalization of Abel’s series, Rend. Circ. Mat. Palermo, 29, 340-346 (1910) · JFM 41.0300.03 |
[42] |
Jensen, J. L.W. V., Sur une identité d’Abel et sur d’autres formules analogues, Acta Math., 26, 307-318 (1902) · JFM 33.0450.01 |
[43] |
C. Kramp; C. Kramp |
[44] |
Lagrange, J. L., Nouvelle méthode pour résoudre les équations littérales par le moyen des séries, (Oeuvres, Vol. 3 (1869), Gauthier-Villars: Gauthier-Villars Paris), 5-73 |
[45] |
Lambert, J. H., Observationes variae in mathesin puram, (Opera Mathematica, Vol. 1 (1946), Orell Füssli: Orell Füssli Zürich), 16-51 |
[46] |
Levine, J.; Dalton, R. E., Minimum periods, modulo \(p\), of first-order Bell exponential integers, Math. Comp., 16, 416-423 (1962) · Zbl 0113.03402 |
[47] |
Manikarnikamma, S. N., Some properties of the series \(∑^∞n = 0 ((n + a)^r x^n\) n!)\), Math. Student, 18, 132-135 (1950) · Zbl 0045.01704 |
[48] |
Moy, A., Problem 2723, solutions by R. Breusch, Amer. Math. Monthly, 86, 788-789 (1979) |
[49] |
Newman, D. J., Problem 4489, solutions by H. F. Sandham, R. Frucht, and M. S. Klamkin, Amer. Math. Monthly, 60, 484-485 (1953) |
[50] |
Pólya, G.; Szegö, G., Aufgaben und Lehrsätze aus der Analysis (1964), Springer-Verlag: Springer-Verlag Berlin, erster Band · JFM 51.0173.01 |
[51] |
Problem. Problem, Math. Sb., 4, 39 (1869) |
[52] |
Ramanujan, S., Collected Papers (1962), Chelsea: Chelsea New York |
[53] |
Ramanujan, S., Notebooks (1957), Tata Institute of Fundamental Research: Tata Institute of Fundamental Research Bombay, 2 Vols · Zbl 0138.24201 |
[54] |
Riordan, J., An Introduction to Combinatorial Analysis (1958), Wiley: Wiley New York · Zbl 0078.00805 |
[55] |
Rogers, K., Solving an exponential equation, Math. Magazine, 53, 26-28 (1980) · Zbl 0432.10006 |
[56] |
Roman, S. M.; Rota, G.-C, The umbral calculus, Advan. in Math., 27, 95-188 (1978) · Zbl 0375.05007 |
[57] |
Rota, G.-C; Kahaner, D.; Odlyzko, A., On the foundations of combinatorial theory. VIII. Finite operator calculus, J. Math. Anal. Appl., 42, 684-760 (1973) · Zbl 0267.05004 |
[58] |
Rota, G.-C; Mullin, R., On the foundations of combinatorial theory, (Harris, B., Graph Theory and Its Applications (1970), Academic Press: Academic Press New York), 167-213 · Zbl 0259.12001 |
[59] |
Rothe, H. A., Formulae de serierum reversione demonstratio universalis signis localibus combinatorioanalyticorum vicariis exhibita, (Dissertation (1793)), Leipzig |
[60] |
Tate, J., A Treatise on Factorial Analysis (1845), Bell: Bell London |
[61] |
Touchard, J., Propriétés arithmétiques de certains nombres récurrents, Ann. Soc. Sci. Bruxelles, A53, 21-31 (1933) · Zbl 0006.29102 |
[62] |
Touchard, J., Nombres exponentiels et nombres de Bernoulli, Canad. J. Math., 8, 305-320 (1956) · Zbl 0071.06105 |
[63] |
Williams, G. T., Numbers generated by the function \(e^{e^{x−1}\) · Zbl 0060.08416 |