# 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
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ‎$‎‎\alpha‎>0‎$‎, ‎$‎‎\beta‎<0‎$‎...
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### “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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 $n=1,2,\ldots$, $z>0$, $0<a<1$,...
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### 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 ...