Confluent hypergeometric function of the second kind - KummerU Formula

Question

I have the following formula for KummerU:

U = gamma(1-b)*KummerMz(a,b,z)/gamma(a-b+1) + gamma(b-1)*z.^(1-b).*KummerMz(a-b+1,2-b,z)/gamma(a)

I am generating it in Excel using a Gamma and KummerM function. But I do not get the desired score of .953742 for a=.25, b=.25, z=.5.

Any input to where the calculation is going wrong?

Gamma(1-.25) = 1.2254; KummerMz(a,b,z) = 1.6487; Gamma(a-b+1)=1; Gamma(b-1) = -4.8341; z^(1-b) = .5946; KummerMz(a-b+1,2-b,z) = 1.3454; Gamma(a) = 3.6256. — Juggler_IN, Aug 06 '19 at 10:45

score 1 · Answer 1 · answered Aug 06 '19 at 11:05

I cannot tell where is your error.

Your expression is $$U(a,b,z)=\frac{\Gamma (1-b) }{\Gamma (a-b+1)}\, _1F_1(a;b;z)+z^{1-b}\frac{ \Gamma (b-1)}{\Gamma (a)} \, _1F_1(a-b+1;2-b;z)$$

For the values you give $(a=b=\frac 14,\,z=\frac 12)$, this reduces to $$\sqrt{e}\,\, \Gamma \left(\frac{3}{4},\frac{1}{2}\right)\approx 0.953742$$

The two numerical values you wrote in comments are correct.

1 Answers1