The Theory of the Boiible Gamma Function. 



267 



^ 27r J(l-e--i^)(l-^--2^)' 



and the determination of the axis of the contour and the number M 

 depends upon the distribution of the points wi, - 1, and (wi + (02). 



By means of this function we obtain asymptotic approximations for 

 summations of the type 



2 2 



when 71 is a very large number and s has any complex value. We can 

 also obtain an asymptotic approximation for the product 



n n (a + mi(Di + m2W2) . 



TTl] = = 



Since Stirling's theorem gives the asymptotic evaluation of ?i !, we 

 obtain, on putting a = 0, an extension of Stirling's theorem to two 

 parometers. We find, as the absolute term, the double Stirling func- 

 tion /32(wi,w2), which is the analogue of the simple Stirling form 

 = x/27r/w. All the double asymptotic expansions involve as 

 their coefficients double Bernoullian functions and numbers. The 

 double Bernoullian function 2Sn(«/(oi,w2) is an algebraic polynomial 

 which satisfies the two difference equations 



f {a + (a) = S4a/o>2) + ^^tii^^/^^, 



f(a + -/(«) = S„ («/«.,) + SVi_{OK) ^ 



and possesses properties exactly analogous to the corresponding simple 

 forms. The theory of this function forms Part I of the memoir. 

 From the contour-integral expression for the double Eiemann f func- 

 tion it is possible to obtain similar expressions for the logarithm of the 

 double gamma function and its derivatives, for the first and second 

 double gamma modular forms, and for the logarithm of the double 

 Stirling function. Under certain restrictions these contour-integrals 

 can be transformed into line-integrals. 



The double gamma function admits of transformation and multipli- 

 cation theories developed in Part lY. By means of the latter theory 

 we may express the double Stirling form as a product of double gamma 

 functions of arguments Jo>i, Jt02, and ^{oii -\- (^2) respectively. There 

 is also a transformation theory for the double gamma modular forms 

 and the double Stirling function. 



The extension of Eaabe's formula for the simple gamma function 

 leads to certain " integral formulae." The integral 



VOL. LXVT. Y 



