4
votes
1answer
158 views

Representation for increasing function on unit square

Do you know any result concerning the representation of functions $f : [0,1] \to [0,1]$ continuous and increasing (with $f(0)=0,f(1)=1)$) as convex combinations of a family of particular functions? ...
1
vote
1answer
215 views

Meaning of $\alpha$ in Laguerre polynomials

I found that generalized Laguerre polynomials are: $$ L_n^{\alpha} (x) = \sum_{i=0}^n (-1)^i {n+\alpha \choose n-i} \frac{x^i}{i!}.$$ However, I wonder what is the meaning of $\alpha$ in this ...
5
votes
1answer
214 views

prove equality with integral and series

I am stuck on one question with integral. Help me please to show that with $n=1$ the following is true $$ ...
0
votes
1answer
66 views

Expansions of Hermite functions

I am wondering if someone knows good references. I am looking for expansions of Hermite functions, which gives connections between rates of decay and smoothness of coefficients. Thank you for your ...
1
vote
0answers
146 views

integral with Bessel function

Let $n$ be half an odd integer, say $n=k+1/2, k \in Z$. Let $q\geq 1$. I would like to calculate (or approximate) the following integral $$ \int_0^{\infty}\left(\sqrt{\frac{\pi}{2}}\cdot 1\cdot 3\cdot ...