8.4.4 Euler's Definition of the Gamma Function The gamma function, just as many of the higher-level functions, has a variety of representations. Among them is another product-type formulation given by...

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Extracted text: 8.4.4 Euler's Definition of the Gamma Function The gamma function, just as many of the higher-level functions, has a variety of representations. Among them is another product-type formulation given by Euler. The most direct way to proceed is to give the proposed mathematical structure and then demonstrate that it produces the equation satisfied by the gamma function, i.e., I'(z + 1) = zT(z). Consider the function u(z) u(2) = (1+±) П (8.125) n=1 386 Difference Equations Therefore (1+ ±)**] z+1 u(z+1) = () II n z +1 1+ z+1 n=1 n (1+ ) П n+1 z + 1 n+z+1 n=1 n -()피 (1+) П z +1 1 n+1 n=1 (1+ ±)* | П (1+ %) n=1 (1+ = (2) (=) I %3D (1+ %) n=1 = zu(z). (8.126) Also, observe that from equation (8.125), we have 1 (i) II (1+±)* (1+ %) e(1) = = 1. (8.127) n=1 Therefore, u(z) as given by equation (8.125) is the gamma function.