Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [special-functions]

This tag is for questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).

Special functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications.

Higher transcendental functions are frequently termed special functions. These functions were studied extensively in the eighteenth and nineteenth centuries-by Gauss, Euler, Abel, Jacobi, Weierstrass, Riemann, Hermite, Poincare, and other leading mathematicians of the day. Although many of the functions that they treated were quite recondite and are no longer of much interest today, others (such as the Riemann zeta function, the gamma function, and elliptic functions) are still intensively studied.

Before asking, please make sure that you define your notation very precisely, as
1. not everybody is familiar with the notation for special functions; and
2. a lot of special functions have different notational conventions, depending on the paper/book.

You might want to first check if the special function you are considering is discussed in Abramowitz and Stegun, the Digital Library of Mathematical Functions, or the Wolfram Functions site.

References:

https://en.wikipedia.org/wiki/Special_functions

https://en.wikipedia.org/wiki/List_of_special_functions_and_eponyms

"Special Functions and Their Applications" by R. Silverman

4626 questions
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Proving a Legendre function using generating function: $\int_{-1}^1 (1-2xt+t^2)^{-1/2}P_n(x)dx=\frac{2t^n}{2n+1}$

I must prove that $\int_{-1}^1 (1-2xt+t^2)^{-1/2}P_n(x)dx=\frac{2t^n}{2n+1}$. I know that the generating function is $(1-2xt+t^2)^{-1/2}=\sum_{n=0}^\infty P_n(x)t^n$. I also know that the orthogonality property when l=m gives $\frac{2}{2n+1}$. I…
Aksel'sRose
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Plot a Floquet solution to Mathieu equation

In wikipedia https://en.wikipedia.org/wiki/Mathieu_function#Floquet_solution I want to know how the Floquet solution is plotted. One way I am thinking is to write Floquet solution in terms of the elemental solutions: Mathieu sin and cos, so that I…
Tim
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sum of integral parts of real number and fraction

For any real x and positive integer n ,show that [x] + [x +1/n] + [x + 2/n] + .... + [x + n-1/n] = [nx] I have used the fact that x-1 < [x] <= x,for all terms and added,but not able to get tight upper bound with it
user3615045
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Where was the mistake

we know that $$\frac{\pi^2}{6}=\int_{0}^{\infty}\frac{t}{e^t-1}dt$$, we also know that $\frac{t}{e^t-1}$ is the generating function for the Bernoulli numbers i.e $ \frac{t}{e^t-1} =\sum_{n=1}^{\infty}\frac{B_nt^n}{n!}$. If we use this generating…
user90533
  • 576
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How can I scale a value when it is within a threshold?

I am not a mathematician so I'm not even sure of the correct language to describe this. I also don't know what appropriate tags are for this question so please amend as necessary. I am looking for a function that would scale it's input when it…
Greg B
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hypergeometric transformation

I came across the following ${}_3F_2$ hypergeometric polynomial: $$ {}_3F_2\left(\left.\begin{array}{c} 1,1,-n\\ 2, -1-2n \end{array}\right| -x\right) $$ for some large $x > 0$. I am wondering if there is some transformation or identity that can…
user58955
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Does anyone know a function that can describe a harmonic series?

I want to find a function that satisfies the following functional equation: $F(z+1)=1/z+F(z)$ This is a generalization of harmonic series 1 + 1/2 + 1/3 + 1/4 + ..., and is similar to the gamma function which satisfies the relation $F(z+1)=zF(z)$. I…
XIN
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Recursive function including Bessel functions

I was wondering if anybody knows how to solve (numerically) the following recursive equation (found in http://dx.doi.org/10.1109/3.250392): $$E^{o}_{k}=\sum^{\infty}_{q=-\infty}J_{q-k}(2m)E^{o}_q,$$ with $J$ being the Bessel function of the first…
Heinrich
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Relation between hypergeometric functions?

Is there any relations between the following hypergeometric functions? $$\ _2F_1(1,-a,1-a,\frac{1}{1-z})$$ $$\ _2F_1(1,-a,1-a,{1-z})$$ $$\ _2F_1(1,a,1+a,\frac{1}{1-z})$$ $$\ _2F_1(1,a,1+a,{1-z})$$
Nahc
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Functions that sum to zero under cyclic index permutations?

Consider $n$ real vectors of arbitrary dimension $x_i$ with $i\in \{1,2,...,n\}$. Furthermore, consider a function $f(x_1,x_2,x_3)$ where actually the function can only depend on scalars $x_i\cdot x_j$ so that $i,j\in \{1,2,3\}$ count as input…
Kagaratsch
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Proving or disproving that if $\Gamma(a)+\Gamma(b)= 121\,645\,106\,635\,852\,800$ both $a$ and $b$ are integers.

I found some formula about special function very complicated, so I am curious how you people solve this by hand. $$\Gamma(a)+\Gamma(b)= 121\,645\,106\,635\,852\,800$$ but $a$ and $b$ are very small actually, condition: at least one of $a$ and $b$…
Victor
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Function with special behavior near zero

I'm looking for functions that have the following behaviors: $f(x) \to 0$ as $|x| \to 0$, as $x \to 0^+$, $\alpha < \frac{{df(x)}}{{d(x)}}$ for any $0<\alpha<\infty $. One example of this kind of functions is $f(x)=x^a$ where $0
Reza H. Khayyami
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Asymptotic for Bessel Function

We have that, $$J_p(x) = \sqrt{\frac{2}{\pi x}} \sin \left( x - \frac{p\pi}{2} + \frac{\pi}{4}\right) + \frac{r_p(x)}{x\sqrt{x}}$$ We also know that there exists $M>0$ such that $|r_p(x)| \leq M$. I am curious, can you tell me an estimate for $M$?…
Nicolas Bourbaki
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Important Functions That Are Multivariable Integrals

There are lots of "important" functions of one variable that are defined in terms of integrals for which no closed form exists, like the Gamma Function and the normal distribution. Are there any such functions that arise in the multivariable…
J126
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Does a function relationship for a specific $y$ hold for any?

Let $f:\mathbb{R}\times \mathbb{R}\mapsto \mathbb{R}$ , continous $\forall$ $y$ in $\mathbb{R}$, $\exists$ $a,b$ such that $f(x,y) = ax + b$ $\forall$ $x$ in $\mathbb{R}$, $\exists$ $a',b'$ such that $f(x,y) = a'y + b'$ 1/ does $a$,$b$,$a'$,$b'$ are…
Toto
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