Questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).
0
votes
0answers
28 views
Generalization of $e^{t A}$ to $e^{t^{\alpha}A}$
I would like to give a mathematical structure of an operator of the form $(S_{\alpha}(t))_{t \geq 0}$ where $S_{\alpha}(t)=e^{t^{\alpha} A}$ as it was done for the operator $S(t)=e^{t A}$ for ...
2
votes
1answer
38 views
Sum involving the hypergeometric function, power and factorial functions
I am finding some trouble in calculating the following sum involving the hypergeometric function, power and factorial functions.
$$
\sum_{y=1}^\infty \mathrm{e}^z \cdot {}_1F_1\left(1-y;2;-z\right) ...
1
vote
1answer
25 views
Spherical harmonics expansion for a particular function
On the unit sphere, each square-integrable function can be expanded as a linear combination of spherical harmonics :
$$ f(\theta,\phi) = \Sigma_{l=0}^\infty \Sigma_{m=-l}^{+l} f_{lm} Y_{lm} ...
3
votes
0answers
66 views
Identity involving $\zeta(3)$
This is a follow-up of this question and my partial answer to it. I've found that the proof of the 2nd identity reduces to showing that
...
0
votes
0answers
32 views
Solve For a Generating Function
Special Functions: Legendre, Bessel, Elliptic, and Confluent Hypergeometric Functions
Considering the following functions of $f_{n}(x)$ defined by:
$$
(a)---- ...
0
votes
1answer
42 views
Legendre Polynomial as Infinite Series
We've been covering Special Functions such as Legendre Functions, Bessel Functions, and Confluent Hypergeometric Functions
For: $$
f(x)=\left\{\begin{matrix} +1
& 0<x<1\\
-1 & ...
3
votes
2answers
46 views
Proof of $\left| \operatorname{cn}\left( x(1+ i) \mid m \right)\right|=1$ for $m=\frac{1}{2}$ and $x \in \mathbb{R}$
Let $\operatorname{cn}(u\mid m)$ be Gudermann's notation for the Jacobi elliptic function $\operatorname{cn}$. It is well known to be doubly periodic. For $0<m<1$ the two periods, $4 K(m)$ and ...
1
vote
1answer
70 views
Numerical values of the Jacobi elliptic function sn: Wolfram Alpha vs. Maple vs. C++?
I have a problem to check the validity of an algorithm I've implemented in C++ to compute the Jacobi elliptic function $\mathrm{sn}(u, k)$ (inspired and improved from Numerical Recipes 3rd edition). I ...
0
votes
0answers
21 views
$_{1}{{F}_{1}}\left( 1,c,(a+b)x \right)-\, _{1}{{F}_{1}}\left( 1,c,ax \right)=(?)$
I am looking for:
$$_{1}{{F}_{1}}\left( 1,c,(a+b)x \right)-\, _{1}{{F}_{1}}\left( 1,c,ax \right)=(?)$$
where $x$ is between $0$ and $T$ and $a,b,c>0$.
I know the asymptotic behavior as $x$ goes ...
3
votes
1answer
113 views
How to derive the Golden mean by using properties of Gamma function?
The Golden mean known as $\frac{1+\sqrt{5}}{2}$.
How could one show the Golden mean can be expressed as
$$
\frac{2\cdot 3\cdot 7\cdot 8\cdot 12\cdot 13\cdots}{1\cdot 4\cdot 6\cdot 9\cdot 11\cdot ...
1
vote
2answers
35 views
Looking for function with specific properties
I need a function $f$ that is arbitrarly times differentiable and which has integral
$$\int _a^b f(x) dx $$ strictly positive (where $a$ and $b$ are fixed), and for all derivatives, we have ...
4
votes
2answers
64 views
Help with special function differential equation
this is my first time to use this site. Please let me know if the equations are unreadable, latex isn't my first language.
We've been covering Legendre, Bessel, and Confluent Hypergeometric ...
3
votes
3answers
85 views
The number $ \frac{(m)^{(k)}(m)_k}{(1/2)^{(k)} k!}$
For a real number $a$ and a positive integer $k$, denote by $(a)^{(k)}$ the number $a(a+1)\cdots (a+k-1)$ and $(a)_k$ the number
$a(a-1)\cdots (a-k+1)$. Let $m$ be a positive integer $\ge k$. Can ...
1
vote
0answers
40 views
Gamma Function Problem
Hi is it fair to write $$\Gamma(1+ix)=ix\Gamma(ix) $$
and then to say that $\Gamma(ix)=\exp(i\arg(\Gamma(x)))$. If not can anyone explain what $\arg(\Gamma(x))$ is defined as please, I always get ...
1
vote
0answers
19 views
Identity for reducing the power in the parameters of Hypergeometric functions
Is there any identity/formula for reducing/increasing the power in the parameters of the Gauss Hypergeometric function $ _2F_1(a,b;c;z^d)$ (d is a real) let's say to z?
Is there also any identity for ...
