2
votes
1answer
110 views

Integral involving the Spherical Bessel Function of the First Kind

How can I prove the equation below using Spherical Bessel Function Recurrence Relation? (where $ j_{n}(x) $ means Spherical Bessel function of first kind) Definition using BesselJ function: $$ ...
2
votes
0answers
134 views

Special map $f(x_n)=x_{{n+1} [\mod 4]}$

As mentioned in the headline I am considering the map $f(x_n)=x_{{n+1} [\mod 4]}$. It looks a little bit similar to the dyadic (bernoulli) map but instead of $\mod 1$ we have $\mod 4$ which means that ...
2
votes
0answers
58 views

Recurrence relation for polygamma reflection polynomials

In the reflection formula for the polygamma function $$ \psi^{(n)}(1-z) + (-1)^{n+1}\psi^{(n)}(z) % = (-1)^{n} \pi \frac{d^{n}}{d z^{n}} \cot (\pi z) $$ the right hand side is a polynomial ...
0
votes
0answers
138 views

Sum of two Bessel function of first kind

I want to find an expression for the sum of two Bessel functions of first kind with the same argument but a different order, i.e. $F(i,j)=|J_{i+j}(x)+(-1)^j J_{i-j}(x)|^2$. Is there any way of ...
2
votes
1answer
145 views

Recurrence relation in complex domain

I am bumping in the following problem : the expansion of $$x^{i/k} \operatorname{LerchPhi}[x,1,i/k]$$ leads, for each value of i, to a linear combination of k terms, each of them writing $$a(j) ...
2
votes
1answer
75 views

Constant term of recursively defined polynomials related to the Lambert W function

The Lambert $W$ function has the property that $$ W'(x) = \frac{W(x)}{x[1+W(x)]}, $$ and using this one can show that its Taylor expansion about $x=a$ has the form $$ W(x) = W(a) + ...
0
votes
1answer
82 views

Recurrence equation and special functions

Can someone give me a proof or a hint on why the recurrence equation: $$g(k+2)=k*g(k+1)-g(k)$$ has the solution: $$g(k)=c_1 {_0\tilde F_1}(;k;-1)+c_2 Y_{k-1}(2)$$ where ${_0\tilde F_1}(;a;x)$ is the ...
2
votes
1answer
134 views

How can I express such function as known functions or power series?

$$\int_0^x \cfrac{1}{1+\int_0^t \cfrac{1}{2+\int_0^{t_1} \cfrac{1}{3+\int_0^{t_2} \cfrac{1}{\cdots} dt_3} dt_2} dt_1} dt =f(x)$$ $$\int_{0}^{x} \frac{1}{n+h_{n+1}(t)}{d} t=h_n(x)$$ ...
3
votes
0answers
103 views

Perrin numbers in terms of the generalized hypergeometric function?

Given the roots of $x^3=x^2+1$, we have sequence A001609, $M(n) = x_1^n+x_2^n+x_3^n = \,_3F_2\left(\frac{-n}{3}, \frac{1-n}{3}, \frac{2-n}{3};\; \frac{1-n}{2}, \frac{2-n}{2};\; ...
3
votes
1answer
180 views

What's the reasoning for this recurrence on $q$-multinomial coefficients?

I'm familiar with the recurrence for binomial coefficients based on Pascal's triangle. However, in general, there is the recurrence for $q$-multinomial coefficients given by $$ ...
2
votes
2answers
368 views

Representing affine transform of Legendre polynomials

I have a function defined as a set of weighted Legendre polynomials: $f(x)=\alpha_0 P_0(x) + \alpha_1 P_1(x) + \alpha_2 P_2(x) +\ldots$. I have another function similarly defined with Legendre basis ...