A Note on a Family of Alternating Sums of Products of Binomial Numbers

Autor/in	Gauthier, N.
Titel	A Note on a Family of Alternating Sums of Products of Binomial Numbers
Quelle	In: International Journal of Mathematical Education in Science and Technology, 44 (2013) 2, S.253-264 (12 Seiten)Infoseite zur Zeitschrift PDF als Volltext Verfügbarkeit
Sprache	englisch
Dokumenttyp	gedruckt; online; Zeitschriftenaufsatz
ISSN	0020-739X
DOI	10.1080/0020739X.2012.678895
Schlagwörter	Numbers; Algebra; Mathematical Concepts; Mathematical Logic; Validity + Suchen Sie Ihr Suchwort? Zahlenraum; Mathematical logics; Mathematische Logik; Gültigkeit
Abstract	We study the following family of integral-valued alternating sums, where -infinity equal to or less than m equal to or less than infinity and n equal to or greater than 0 are integers [equation omitted]. We first consider h[subscript m](n) for m and n non-negative integers and show that it is of the form 2[superscript n + 2m] - P[subscript m](n), where P[subscript m]n may be represented as a polynomial of degree m in n, or expressed as a non-polynomial closed form given by a sum of binomial numbers. We then consider h[subscript m]n for m = -ImI a negative integer and for n a non-negative integer. This reveals, in particular, that h[subscript-ImI]n = 0 for 0 equal to or less than n equal to or less than ImI, that h[subscript-ImI](n) = 2[superscript n-2ImI] for n equal to or greater than 2ImI. We also show that h[subscript-ImI](n + ImI) is a polynomial of degree n - 1 in ImI, for fixed n equal to or greater than I, with ImI equal to or greater than I, and we give expressions for the coefficients. (As Provided).
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Update	2017/4/10