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 | MEDIUM | LARGE Below is the OCR-scanned text from this page: 23. Bernoulli and Euler Polynomials-Riemann Zeta 23.1.2 B,=B,(O) n=o, 1, . . . 1 1 1 23.1.3 Bo=l, Biz--- BZ=-J B4=-- 2' 6 30 Function E,=~"E, (;)= integer Eo= 1 , E,= - 1 , E4=5 Mathematical Properties 23.1. Bernoulli and Euler Polynomials and the Euler-Maclaurin Formula Generating Functions n.=1, 2, . . . n=O, 1, . . . teZt t" 23.1.1 ---=C B,(z) - e'-1 n = ~ n! Eb(x)=nEn-l(z) En(~+l)+En(~)=22" Bernoulli and Euler Numbers n=O, 1, . . . n=ol 1,. . . En(l-~)=(-l)~En(x) (- l)"+'En( - 2) =En (2) - 22" Itl<* n=O, 1 , . . . (For occurrence of B, and En in series expansions of circular functions, see chapter 4.) Sums of Powers 23.1.5 Bb(s)=nB,-,(z) 23.1 -6 B,(z f 1) - Bn(X) =W"-' 23.1.7 k=O 23.1.8 Bn(l-~)=(-l)"Bn(~) 23.1.9 (-I)"Bn(-z)= Bn(z)+mn-l 23.1.10 m-1 k=O B,(mz)=m"-' m=1,2,. . . n=1,2,. . . n=O, 1 , . . . n=O1 1 , . . . n=O, 1 , . . . n=O, 1 , . . . n=O, 1 , . . . n=O, 1 , . . . m=1,3,. . . m-1 2 n f l k=O 804 n=O, 1 , . . . m=2, 4,. . . 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. ©2000 ConvertIt.com, Inc. All rights reserved. Terms of Use.