Questions tagged [special-functions]

Questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).

3,460 questions
82 views

33 views

Putting some polynomials in terms of orthogonal polynomials

I came across some polynomials which take the form $$F_n(x)=\sum_{k=0}^{\lfloor n/2\rfloor}(-1)^k\binom{2k}{k}\binom{n-k}{k}x^{n-2k}.$$ I noticed that these look pretty similar to the series for the ...
26 views

Can a logarithm function with two variable be expressed as a Meijer-G function?

Let {x},{y}>0 and {a},{b}>0. can the function log(1+ax+by) be expressed as a Meijer-G function?
34 views

Spherical expansion of an exponential function?

We know that a normal planewave can be Rayleigh expanded by spherical harmonics as$$e^{i\vec k·\vec r}=4π\sum_{l=0}^∞\sum_{m=-l}^li^lj_l(kr)Y_{lm}(\hat{\vec k})Y_{lm}^*(\hat{\vec r}).$$ Does any body ...
42 views

42 views

46 views

50 views

A Continuous function $f: \overline{B_1(0)} \subset \ell^2\to \mathbb{R}$ which does not reach the maximum?

If necessary, recall that $$\ell^2 = \{x=\{x_n\}_n\subset \mathbb{R} : \|x\|^2:=\sum_n |x|^2<\infty\}$$ and $\overline{B_1(0)}$ is the closed unit ball with respect to that norm. Can we ...
71 views

Calculate sum of series ‎$‎\sum_{k=1}^\infty (k+x-m)^\alpha e^{\beta (k+x-m)}$‎.

‎I've been stuck with calculating the sum of series of the following problem. Can you help me?‎ ‎ $‎\sum_{k=1}^\infty‎(k+x-m)^\alpha ‎e^{‎\beta‎(k+x-m)}‎$ for ‎$‎‎\alpha‎>0‎$‎, ‎$‎‎\beta‎<0‎$‎...
420 views

“Continuized” Taylor Series? $\sin(x)=\sum \frac{(-1)^nx^{2n+1}}{(2n+1)!}=\int_{-1}^\infty \frac{\cos(\pi n) x^{2n+1}}{G(2n+1)}dn$?

~~not trying to reinvent the Laplace transform, but just an exploration into these particular series and integrals~~ Current answers don't fully address the 5 questions, so any new ideas or ...
34 views

A conditional expectation of the beta binomial distribution?

Consider a beta binomial distribution where the number of trials, $n$, is odd and the shape parameters of the underlying beta distribution, $\alpha$ and $\beta$, are equal. Is there a closed form ...
92 views

Real Analysis Methodologies to Prove $-\int_x^\infty \frac{\cos(t)}{t}\,dt=\gamma+\log(x)+\int_0^x \frac{\cos(t)-1}{t}\,dt$ for $x>0$

In this question, the OP asked to prove the Cosine Integral identity $-\int_x^\infty \frac{\cos(t)}{t}\,dt=\gamma+\log(x)+\int_0^x \frac{\cos(t)-1}{t}\,dt$, where $\gamma$ is the Euler-Mascheroni ...
77 views

Why are there two different recurrences for Gegenbauer polynomials?

As I mentioned previously, I've been reading up on Gegenbauer polynomials in preparation for a blog post on the kissing number problem—specifically, the Delsarte method. To make a long story short, ...
44 views

23 views

Sum involving hypergeometric 2F2 function

I'm trying to simplify the following sum: $$\sum_{i=0}^n\frac{z^i}{(n-i)!}\,\frac{1}{(1+a)_i\,(1-a)_i}\sum_{j=0}^i(-1+a)_j\,(-1-a)_j\frac{(-z)^j}{j!},$$ where $n=1,2,\ldots$, $z>0$, $0<a<1$,...
15 views

Integrate a Generalized incomplete gamma function

My question is about an integral but not an ordinary one $\int_{le^{-bt'}}^{l}s^{l-1}e^{-s}ds$ I have the idea of use the leibniz integral rule because I don't want to use the Generalized ...
This question is closely related to An integral involving a Gaussian and an Owen's T function. and An integral involving error functions and a Gaussian . Let $\nu_1 \ge 1$ and $\nu_2 \ge 1$ be ...