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The confluent hypergeometric function of the first kind _1F_1(a;b;z) is a degenerate form of the hypergeometric function _2F_1(a,b;c;z) which arises as a solution the ...

The confluent hypergeometric function of the second kind gives the second linearly independent solution to the confluent hypergeometric differential equation. It is also ...

A generalization of the confluent hypergeometric differential equation given by (1) The solutions are given by y_1 = x^(-A)e^(-f(x))_1F_1(a;b;h(x)) (2) y_2 = ...

A reduction system is called confluent (or globally confluent) if, for all x, u, and w such that x->_*u and x->_*w, there exists a z such that u->_*z and w->_*z. A reduction ...

The second-order ordinary differential equation xy^('')+(c-x)y^'-ay=0, sometimes also called Kummer's differential equation (Slater 1960, p. 2; Zwillinger 1997, p. 124). It ...

_0F_1(;a;z)=lim_(q->infty)_1F_1(q;a;z/q). (1) It has a series expansion _0F_1(;a;z)=sum_(n=0)^infty(z^n)/((a)_nn!) (2) and satisfies z(d^2y)/(dz^2)+a(dy)/(dz)-y=0. (3) It is ...

Special functions which arise as solutions to second order ordinary differential equations are commonly said to be "of the first kind" if they are nonsingular at the origin, ...

Special functions which arise as solutions to second order ordinary differential equations are commonly said to be "of the first kind" if they are nonsingular at the origin, ...

Another name for the confluent hypergeometric function of the second kind, defined by where Gamma(x) is the gamma function and _1F_1(a;b;z) is the confluent hypergeometric ...

rho_n(nu,x)=((1+nu-n)_n)/(sqrt(n!x^n))_1F_1(-n;1+nu-n;x), where (a)_n is a Pochhammer symbol and _1F_1(a;b;z) is a confluent hypergeometric function of the first kind.