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General Confluent Hypergeometric Differential Equation

A generalization of the confluent hypergeometric differential equation given by

y^('')+((2A)/x+2f^'+(bh^')/h-h^'-(h^(''))/(h^'))y^'+[((bh^')/h-h^'-(h^(''))/(h^'))(A/x+f^')+(A(A-1))/(x^2)+(2Af^')/x+f^('')+f^('2)-(ah^('2))/h]y=0.
(1)

The solutions are given by

y_1=x^(-A)e^(-f(x))_1F_1(a;b;h(x))
(2)
y_2=x^(-A)e^(-f(x))U(a,b,h(x)),
(3)

where _1F_1(a;b;z) is a confluent hypergeometric function of the first kind and U(a,b,z) is a confluent hypergeometric function of the second kind (Abramowitz and Stegun 1972, p. 505).

See also

Confluent Hypergeometric Differential Equation

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 505, 1972.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 123, 1997.

Referenced on Wolfram|Alpha

General Confluent Hypergeometric Differential Equation

Cite this as:

Weisstein, Eric W. "General Confluent Hypergeometric Differential Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GeneralConfluentHypergeometricDifferentialEquation.html

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