Ward’s Generalized Special Functions

Anthony G. Shannon^1, and Ömür Deveci²

¹Emeritus Professor, University of Technology Sydney, NSW 2007, Australia, Fellow, Warrane College, University of New South Wales, Kensington NSW 2033

²Department of Mathematics, Faculty of Science & Letters, Kafkas University, 36100, Kars, Turkey

Cite this paper:
Anthony G. Shannon and Ömür Deveci. Ward’s Generalized Special Functions. Turkish Journal of Analysis and Number Theory. 2018; 6(3):90-92. doi: 10.12691/tjant-6-3-4

Abstract

This paper considers generalizations of Bernoulli and Euler numbers to clarify and extend some known relations studied by Morgan Ward. It does this with the Euler-Maclaurin sum formula. It relates the mappings to category theory as a means of applying the ideas further.

Keywords:
Special function Bernoulli polynomial Euler function difference operator normal sequence divisibility sequence difference operator generalized integer Fermatian numbers Euler-Maclaurin sum-formula commutative diagram category

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References:

[1]	Ward, Morgan, “Divisibility sequences,” Bulletin of the American Mathematical Society, 42, 843-8455, 1936.
[2]	Ward, Morgan, “Memoir on elliptic divisibility sequences,” American Journal of Mathematics, 70 (1), 31-74, 1948.
[3]	Gould, H.W., “The bracket-function and Fontené-Ward generalized binomial coefficients with applications to fibonomial coefficients,” The Fibonacci Quarterly, 7 (1), 23-40, 55, 1969.
[4]	Ward, Morgan. “A calculus of sequences,” American Journal of Mathematics, 58, 255-266, 1936.
[5]	Cohen, G.L., “Selberg formulae for Gaussian integers,” Acta Arithmetica, 16, 385-400, 1975.
[6]	Dickson, L.E. History of the Theory of Numbers, Volume 1. New York, Chelsea, Ch.XVI, 1952.
[7]	Carlitz, L. and Riordan, J., “Two element lattice permutation numbers and their q-generalization,” Duke Mathematical Journal, 31, 371-388, 1964.
[8]	Shannon, A.G., “Some Fermatian Special Functions,” Notes on Number Theory and Discrete Mathematics, 9 (4), 73-82, 2003.
[9]	Hoggatt, Verner E. Jr., “Fibonacci numbers and generalized binomial coefficients,” The Fibonacci Quarterly, 5, 383-400, 1967.
[10]	Rado, R., “A new proof of a theorem of v. Staudt,” Journal of the London Mathematical Society, 9 (1), 85-88, 1934.
[11]	Rado, R., “A note on the Bernoullian numbers,” Journal of the London Mathematical Society, 9 (1), 88-90, 1934.
[12]	Carlitz, L., “The Staudt-Clausen theorem,” Mathematics Magazine, 35, 131-146, 1961.
[13]	Horadam, A.F. and Shannon, A.G.,“Ward’s Staudt-Clausen problem,” Mathematica Scandinavica, 29 (4), 239-250, 1976.
[14]	Roman, Steven, The Umbral Calculus, Orlando, FL: Academic Press, Ch.6, 1984.
[15]	Allen, A.L. and Shannon, A.G., “Note on a category model for an abstraction mechanism.” Psychological Reports. 27: 591-594, 1970.