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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: mERGEOMETRIC F'UNCI'IONS 557 15.1.21 (&++b+WO, -1, -2,. * .I 15.1.25 F(u, b; +U+3b+ 1; 4) =2+(~-- b) -'r (1 + 4 ~ + 3b) I (\$4 r (3+ I-I-- [r (3+ 3 4 r (44 I-' I (~(a+b)+l#O, -1, -2,. . .) 15.1.26 F(u, 1 -a; 6 ; 4) = 2I-%*r(b) [r(\$a+\$b) r (+++b-4~)]-~ ( b z 0 , -1, -2,. . .) 15.1.27 F(1, 1; a+l; +)==a[+(*+3a)-+(3aa>l (a#-1, -2, -3 , . . .) 15.1.28 w, a; a+l; 4)=2"-'a[+(-i+W-+(+U)l (a#-1, -2, - 3 , . . .) (*-2U#O, -1, -2,. . .) 15.1.31 F (a, +a++; +a++; 15.2. Differentiation Formulas and Gauss' Relations for Contiguous Functions Differentiation Formulas d ab - F(u, b; C; z)=- F ( u + ~ , b + l ; ~ + 1 ; Z ) dz C 15.2.1 15.2.2 - dn ~ ( a , b; c; z)== F(a+n, b+n; c+n; z) dz" (c) n 15.2.3 dz" dn [f+n-lF(a, b; c; z)]=(~),Za-~F(a+n, 6; c; z ) 15.2.4 F(a, b; c-n; z) dn - [Z~-'F(U, b; C; Z)]=(C-~),Z~-"-' dz" 15.2.5 - dn [Zc-a+n-l dz" (l-z)"+"-"F(U, b; c; z ) ] = (C-a),ZC-a-1 (l-z)a+a-c-nF(a-n, b; c; z) 15.2.6 dn dzn - [ (l-Z)"+*--"F(a, b; c; z)] (1 -Z)a++-C-nF (a, b; c+n; z> 3 (c-a) n (c--b) n (c) n 15.2.7 dn - [ (1 -Z)"+"-'F(a, 6 ; c; z ) ] d zn Gauss' Relations for Contiguous Functions The six functions F ( a f 1, b; c; z), F(a, b f 1; c; z), F(a, b; c f l ; z) are called contiguous to F(u, b; c; z). Relations between P(a, b; c; z) and 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.