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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: 558 HYPERGEOMETRIC FUNCTIONS any two contiguous functions have been given by Gauss. By repeated application of these rela- tions the function F(a+m, b+n; c+l; z) with integral m, n, Z(c+l#O, -1, -2, . . .) can be expressed as a linear combination of F(a, b; c; z) and one of its contiguous functions with coefficients which are rational functions of a, b, c, z. 15.2.10 (c-u)F(a-1, b; c; z)+(2a-c--az+bz)F(a, b; c; 2) +a(z-l)F(a+l, b; c; z)=O 15.2.11 (c-b)F(a, b-1; c; 2)s (2b-c-bz+az)F(a, b; c; 2) +b(z-l)F(a, b+l; c; z)=O 15.2.12 c(c-l)(z-l)F(a, b; c-1; 2) +c[c-1-(2c--a-b-l)z]F(a, b; c; 2) +(c-a)(c-b)zF(a, b; c+l; z)=O 15.2.13 [c-2a-(b--u)z]F(a, b; c; 2) +a(l-2)F(a+l, b; c; 2 ) -(c-a)F(a-1, b; c; z)=O 15.2.14 (b-u)F(a, b; c; z)+aF(a+l, b; c; 2) -bF(a, b+l; c; z)=O 15.2.15 (c-a-b)F(a, b; c; z)+a(l-z)F(a+l, 4; c; 2) -(C-b)F(a, b-1; c; z)=O c[a-(c-b)z]F(a, b; c; 2)-m(l-z)F(a+l, b; c; 2) +(c-a)(c-b)zF(a, b; c+l; z)=O 15.2.16 15.2.17 (c-a-l)F(a, b; c; z)+aF(a+l, b; c; 2) - (c- l)F(a, b; c- 1; 2) =o 15.2.18 (c-a-b)F(a, b; c; 2)-(c-u)F(a-1, b; c; 2) +b(l-z)F(a, b+l; c; z)=O 15.2.19 (b-u)(l-z)F(u, b; c; 2)-(c-u)F(a-1, b; c; 2) +(c-b)F(a, b-1; c; 2) =o 15.2.20 C(l-Z)F(U, b; c; 2)-cF(a-1, b; c; 2) +(c-b)zF(a, b; c+l; z)=O *See page II. 15.2.21 [a-1-(c-b-l)z]F(a, b; c; 2) +(c-a)F(a-1, b; c; 2) -(c-l)(l-z)F(u, b; c-1; z)=O 15.2.22 [c--2b+(b--a)z]F(u, b; c; 2) +b(l-z)F(a, b+l; c; 2 ) -(C-b)F(a, b-1; c; z)=O 15.2.23 c[b-(c-a)z]F(a, b; c; 2)-bc(1-z)F(a, b+l; c; 2) +(c-a)(c-b)zF(a, b; c+l; z)=O 15.2.24 (c-b-l)F(a, b; c; z)+bF(a, b+l; c; 2) -(c-l)F(a, b; c-1; z)=O 15.2.25 c(1-zz;F(a, b; c; 2)-cF(a, b-1; c; 2) It + (c-a)xF(a, b; c+l; z)=O 15.2.26 [b-1-(c-a-l)z]F(a, b; c; 2) +(c-b)F(a, b-1; c; 2) -(c-l)(l-z)F(u, b; c-1; z)=O 15.2.27 C[C-l-(2c--a-~-l)z]F(a, b; c; 2) +(c-a)(c-b)zF(a, b; c+l; 2) -c(c-l)(l-z)F(u, b; c-1; z)=O 15.3. Integral Representations and Transfor- mation Formulas Integral Representations 15.3.1 F(a, b; c; 2)= (9c>Wb>O) The integral represents a one valued analytic func- tion in the zplane cut along the real axis from 1 to OJ and hence 15.3.1 gives the analytic continua- tion of 15.1.1, F(a, b; c; z). Another integral representation is in the form of a Mellin-Barnes integral 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.