13 Confluent Hypergeometric FunctionsKummer Functions13.4 Integral Representations13.6 Relations to Other Functions

If $a,b\in \mathrm{\u2102}$ such that $a\ne -1,-2,-3,\mathrm{\dots}$, and $a-b\ne 0,1,2,\mathrm{\dots}$, then

13.5.1 | $$\frac{M\left(a,b,z\right)}{M\left(a+1,b+1,z\right)}=1+\frac{{u}_{1}z}{1+}\frac{{u}_{2}z}{1+}\mathrm{\cdots},$$ | ||

where

13.5.2 | ${u}_{2n+1}$ | $={\displaystyle \frac{a-b-n}{\left(b+2n\right)\left(b+2n+1\right)}}$, | ||

${u}_{2n}$ | $={\displaystyle \frac{a+n}{\left(b+2n-1\right)\left(b+2n\right)}}$. | |||

This continued fraction converges to the meromorphic function of $z$ on the left-hand side everywhere in $\mathrm{\u2102}$. For more details on how a continued fraction converges to a meromorphic function see Jones and Thron (1980).

If $a,b\in \mathrm{\u2102}$ such that $a\ne 0,-1,-2,\mathrm{\dots}$, and $b-a\ne 2,3,4,\mathrm{\dots}$, then

13.5.3 | $$\frac{U\left(a,b,z\right)}{U\left(a,b-1,z\right)}=1+\frac{{v}_{1}/z}{1+}\frac{{v}_{2}/z}{1+}\mathrm{\cdots},$$ | ||

where

13.5.4 | ${v}_{2n+1}$ | $=a+n$, | ||

${v}_{2n}$ | $=a-b+n+1$. | |||

This continued fraction converges to the meromorphic function of $z$ on the left-hand side throughout the sector $$.

See also Cuyt et al. (2008, pp. 322–330).