![E(p;alpha_r:rho_s:x)=(Gamma(alpha_(q+1)))/(Gamma(rho_1-alpha_1)Gamma(rho_2-alpha_2)...Gamma(rho_q-alpha_q))product_(mu=1)^qint_0^inftylambda_mu^(rho_mu-alpha_mu-1)(1+lambda_mu)^(-rho_mu)dlambda_muproduct_(nu=2)^(p-q-1)int_0^inftye^(-lambda_(q+nu))lambda_(q+nu)^(alpha_(q+nu)-1)dlambda_(q+nu)int_0^inftye^(-lambda_p)lambda_p^(alpha_p-1)[1+(lambda_(q+2)lambda_(q+3)...lambda_p)/((1+lambda_1)...(1+lambda_q)x)]^(-alpha_q+1)dlambda_p,](/images/equations/MacRobertsE-Function/NumberedEquation1.svg) |
where 
is the gamma function and other details are discussed
by Gradshteyn and Ryzhik (2000).
MacRobert's E-Function
See also
Fox H-Function, Kampé de Fériet Function, Meijer G-FunctionExplore with Wolfram|Alpha
References
Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "Definition of the E-Function." §5.2 in Higher Transcendental Functions, Vol. 1. New York: Krieger, pp. 203-206, 1981.Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 896-903 and 1071-1072, 2000.MacRobert, T. M. "Induction Proofs of the Relations between Certain Asymptotic Expansions and Corresponding Generalised Hypergeometric Series." Proc. Roy. Soc. Edinburgh 58, 1-13, 1937-38.MacRobert, T. M. "Some Formulæ for the
-Function." Philos. Mag. 31, 254-260, 1941.Referenced on Wolfram|Alpha
MacRobert's E-FunctionCite this as:
Weisstein, Eric W. "MacRobert's E-Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MacRobertsE-Function.html