Reference to the continued fractions of the form $\mathop{\text{K}}_{n=1}^{\infty}\frac{an+b}{cn+d}$

Question

With some heuristic computation, I conjecture that the generalized continued fraction (CGD)

$$ \mathop{\Large\text{K}}_{n=1}^{\infty}\frac{an+b}{cn+d} = \dfrac{a+b}{(c+d)+\dfrac{2a+b}{(2c+d)+\dfrac{3a+b}{(3c+d)+\ldots}}} $$

written in Gasus' Kettenbruch notation, has the following form

$$ \mathop{\Large\text{K}}_{n=1}^{\infty}\frac{an+b}{cn+d} = \frac{ \frac{a+b}{c} \beta {}_1F_{1}(\alpha, \beta; \gamma)}{\beta (\beta-1) {}_1F_{1}(\alpha, \beta; \gamma) + \alpha \gamma {}_1F_{1}(\alpha+1, \beta+1; \gamma) } $$

where $\alpha = \frac{a}{c^2} + \frac{d}{c} - \frac{b}{a} $, $\beta = \frac{a}{c^2} + \frac{d}{c} + 2$, and $\gamma = -\frac{a}{c^2}$, and ${}_1F_{1}$ is the Kummer's confluent hypergeometric function.

Given the simplicity of the method I applied, I am fairly certain that this type of answer has already been known in literature, although I was unable to find it with a quick googling. So I would like to seek for a help for identifying references to this type of result. I am by no means an expert of this field, and was only mildly motivated by the project Ramanujan Machine which seeks to discover novel type of GCF identities experimentally.

In short, any help for identifying references would be greatly appreciated!

Answer 1

An easier-to-read form is obtained if one starts at $n=0$ instead of $n=1$: $${\raise{-1ex}\mathop{\huge\text{K}}_{n=0}^{\infty}}\frac{an+b}{cn+d}=\frac{a}{c}\frac{{_1F_1}'(\alpha;\beta;\gamma)}{_1F_1(\alpha;\beta;\gamma)}=\frac{a}{c}\frac{\alpha}{\beta}\frac{_1F_1(\alpha+1;\beta+1;\gamma)}{_1F_1(\alpha;\beta;\gamma)},\\\alpha:=\frac{b}{a},\quad\beta:=\frac{a}{c^2}+\frac{d}{c},\quad\gamma:=\frac{a}{c^2}.\quad\color{LightGray}{\left[\frac{a}{c}\frac{\alpha}{\beta}=\cfrac{b}{d+\cfrac{a}{c}}\right]}$$ It can be deduced from one of the recurrences for $_1F_1$, and there are other approaches (I did it myself some time ago, via ODEs for exponential generating functions for the convergents of the CF, which transform into Kummer's ODE for $_1F_1$ after a linear change of variable). In fact the closely related form $$\frac{_1F_1(a+1;b+1;z)}{_1F_1(a;b;z)}=\frac{b}{b-z}{\vphantom{1}\atop+}\frac{a+1}{b+1-z}{\vphantom{1}\atop+}\frac{a+2}{b+2-z}{\vphantom{1}\atop{+\ldots}}$$ does appear in literature (say, in Jones & Thron section $7.3.3$). It may well be known since Kummer, and certainly known to Nørlund who found an $_2F_1$ analogue (here is the idea, the "ODE for EGF" way).