Talk:Confluent hypergeometric function

Whittaker function
Is U(a,b,z) the Whittaker function? (anon, Oct 2006)


 * I don't know, that's not what A&S calls them.linas 00:43, 11 December 2006 (UTC)

I am certainly not an expert, but I now know a bit about Kummer/Whittaker functions. Enough to find severe discrepancies between A&S and maple. Anybody have an opinion about whether I should tack some things up on the main page?

Kummer's function
I am interested in the real part of Kummmer's function in the case a=2n+1, b=a+1 (real part of incomplete gamma). From a numerical point of view, which is cheaper to approximate, what is the convergence like for each and what methods are used? (anon, Nov 2006)

a sub n is defined in this article, but what is b sub n? — Preceding unsigned comment added by 213.122.105.23 (talk) 12:11, 29 April 2015 (UTC)

continuous fraction for $ez$
The [#Application_to_continued_fractions original text] used to say  by setting $b = 0$ and $c = 1$  It is hard to tell what it meant because there was no $c$ around.

$M(1, 2, z)/M(0, 1, z) <BR >= 1/ <BR >1 &minus; 1/2 <VAR >z</VAR >/ <BR >1 + 1/6 <VAR >z</VAR >/ <BR >1 &minus; 2/12 <VAR >z</VAR >/ <BR >1 + 2/20 <VAR >z</VAR >/ <BR >&hellip; <BR >1 &minus; <VAR >k</VAR >/(2 <VAR >k</VAR > &minus; 1) (2 <VAR >k</VAR >) <VAR >z</VAR >/ <BR >1 + <VAR >k</VAR >/(2 <VAR >k</VAR >) (2 <VAR >k</VAR > + 1) <VAR >z</VAR >/ <BR >&hellip; <BR >= 1 + 1/ 1 &minus; 1/2 <VAR >z</VAR >/ <BR >1 + 1/6 <VAR >z</VAR >/ <BR >1 &minus; 1/6 <VAR >z</VAR >/ <BR >1 + 1/10 <VAR >z</VAR >/ <BR >&hellip; <BR >1 &minus; 1/2 (2 <VAR >k</VAR > &minus; 1) <VAR >z</VAR >/ <BR >1 + 1/2 (2 <VAR >k</VAR > + 1) <VAR >z</VAR >/ <BR >&hellip;$

Transforming this fraction with the sequence $(1, 2, 3, 2, &hellip;, 2 <VAR >k</VAR > + 1, 2, &hellip;)$ gives

$1/ <BR >1 &minus; <VAR >z</VAR >/ <BR >2 + <VAR >z</VAR >/ <BR >3 &minus; <VAR >z</VAR >/ <BR >2 + <VAR >z</VAR >/ <BR >&hellip; <BR >(2 <VAR >k</VAR > &minus; 1) &minus; <VAR >z</VAR >/ <BR >2 + <VAR >z</VAR >/ <BR >&hellip; <BR > = (e<SUP ><VAR >z</VAR ></SUP > &minus; 1)/<VAR >z</VAR >$

which is not quite what was postulated.

--Yecril (talk) 13:47, 3 October 2008 (UTC)

Formal power series?
The following is simply too cryptic for inclusion as it stands
 * Moreover,
 * $$U(a,b,z)=z^{-a} \cdot \, _2F_0\left(a,1+a-b;\, ;-\frac 1 z\right),$$
 * where the hypergeometric series $$_2F_0(\cdot, \cdot; ;z)$$ degenerates to a formal power series in z (which converges nowhere).

Please explain precisely what it is that this is supposed to convey, including a reference. Sławomir Biały (talk) 18:37, 3 July 2009 (UTC)


 * Addendum: Presumably this is supposed to hold as an asymptotic series as z&rarr;0 in the right half-plane. But a reference (or at least a clarification) is needed to establish this. Sławomir Biały (talk) 19:05, 3 July 2009 (UTC)

Referring to @book{andrews2000special, title={Special functions}, author={Andrews, G.E. and Askey, R. and Roy, R.}, year={2000}, publisher={Cambridge Univ Pr} } Page 189 They agree, the formal form above diverges and they provide a convergent alternative solution by taking limits on 2F1.
 * $$\frac 1 {\Gamma(a)} \int_{0}^{\infty} e^{-xt} t^{a-1} (1+t)^{b-a-1} dt,$$

Rrogers314 (talk) 20:53, 16 July 2009 (UTC)


 * No one is disagreeing that the "formal form" diverges. The question is, what exactly is intended by the string of symbols
 * $$U(a,b,z)=z^{-a} \cdot \, _2F_0\left(a,1+a-b;\, ;-\frac 1 z\right).$$
 * Because a power series it most certainly is not. Sławomir Biały (talk) 03:03, 21 July 2009 (UTC)

It's the result of various transformations and limits giving a asymptotic series for x "large". The above reference covers this and computes R_n(x) as O(1/x^n). If you would like I could try to capture the reasoning or result. To give credit; how much can I quote before violating copyright? The book is succinct and I have a tendency to wander off; this means that quoting is probably preferred in some instatnces. My guess about your request is:

1) How does this form, both as symbols and series, come about

2) The effectiveness as a asymptotic series.

3) Skipping the actual intermediate details

?? Rrogers314 (talk) 15:17, 18 August 2009 (UTC)

Clarification request
This section seems confusing

<BLOCKQUOTE > ... .Similarly
 * $$U(a,2a,x)= \frac{e^\frac x 2}{\sqrt \pi} x^{\tfrac 1 2 -a} K_{a-\tfrac 1 2} \left(\tfrac x 2 \right),$$
 * When $a$ is a non-positive integer, this equals $$2^{-a}\theta_{-a}\left(\tfrac x 2 \right)$$ where $θ$ is a Bessel polynomial.

</BLOCKQUOTE >

It isn't clear what the function $$ K_{a-\tfrac 1 2} \left(\tfrac x 2 \right) $$ is supposed to be. The preceding text would incline me guess at Kelvin function, but it really shouldn't have to be a guess. Could somebody please add an appropriate definition? Thank you.

I am pretty sure its the modified Bessel function $$K_{v}(x)$$ https://en.wikipedia.org/wiki/Bessel_function#Modified_Bessel_functions:_I%CE%B1_,_K%CE%B1 Rrogers314 (talk) 14:35, 9 December 2017 (UTC)

Multiplication theorem
It's obvious the written equation is wrong. The left-hand side doesn't contain t and the right-hand side doesn't seem to match DLMF. I will wait for other comments/references or edits before changing it though. Perhaps the author had some other formula completely in mind? [] — Preceding unsigned comment added by Rrogers314 (talk • contribs) 14:40, 9 December 2017 (UTC)