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 New Online Book! Handbook of Mathematical Functions (AMS55) Conversion & Calculation Home >> Reference Information Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55) Purchase the electronic edition of this book in Adobe PDF format! FIRST | PREVIOUS | NEXT | LAST | CONTENTS | PAGE | ABOUT | SMALL | MEDIUM | LARGE Below is the OCR-scanned text from this page: 24. Cornbinatorial Analysis Mathematical Properties In each sub-section of this chapter we use a fixed format wliicli elnpliasizes tlic use and rnetliotls of extending the accompanying tables. The format follows this foriii: I. Definitions A. C'ombinatorial B. Generating functions C'. Closed form A. Recurrences B. Checks in computing C. Basic use in numerical analysis In general the notations used are standard. This includes the difference operator A defined on functions of x by Af(z)=f(r+ 1)-f(z), An+Y(r) =A(Anf(z)), the Kronecker delta 6,,, tlie Riemanri zeta function {(s) and the greatest common divisor symbol (m, n ) . The range of the summands for a summation sign without limits is esplained to the riglit of the formula. The notations which are not standard are those for the multinomials which tire arbitrary short- hand for use in this chapter, and those for the Stirling numbers which have never been stand- ardized. A short table of various notations for these numbers follows : 11. Relations 111. Asymptotic and Special Values Notations for the Stirling Numbers Reference First Kind Second Kind This chapter Sf' SLm' (24.21 Fort #p' yy) * I24.71 Jordan S.- e? * [24.10] hloser and Wyman S: 0.- [24.15] Riordan s(n, m) S(n, m) [24.9] Milne-Thomson ( - ' ) R2\$ (;) BL:' m- 1 [24.1] Carlitz ( - l ) n - m S l ( n - l , n - m ) S Z ( ~ , n-m) (24.31 Could Miksa S ( n - m f l , n) ms. (Unpublished tables) (24.171 Gupta u(n, m) We feel that a capital S is natural for Stirling numbers of the first kind; i t is infrequently used for other notation in this contest. But once it is used we have difficulty finding a suitable symbol for Stirling numbers of the second kind. The numbers are sufficiently important to warrant a special and easily recognizable symbol, and yet that synibol iniist be easy to write. We have settled 011 a script capital 3 without any certainty that we liave settled this question pernianently. We feel that the subscript-superscript notation emphasizes tlie generating functions (wliicli are p6wers of mutually inverse functions) from which most of the important relations flow. 24.1. Basic Numbers 24.1.1 Binomial Coefficients I. Definitions A. (2) is the number of ways of choosing m objects froin a collection of n distinct objects without regard to order. B. Generating functions n=O,l,. . . m=O ( 1 - 2) -m- 1 = n=m C. Closed form n2 m - n(n-1). . . (,n-m+l> m! - - - 11. Relations A. Recurrences n+.l ( m )=(z)+(mnl> n>m>l B. Checks 2 111 =o ( - l ) n - m ( ; ) = ( ' ; ' ) ( : ) = ( n o ) [ " 1 ) . - . . (Iiiodp) pi1 prime ma tnl r+s2n a22 *Sw pngr 11. The page scan image above, and the text in the text box above, are contributions of the National Institute of Standards and Technology that are not subject to copyright in the United States.