# Questions tagged [bessel-functions]

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### Convolution of two modified Bessel functions

Does a closed formula (or power series expansion) for the following convolution exist? $$I_{\nu}(x)=\int_{0}^{\infty} K_{\nu}(x-\tau)K_{\nu}(\tau)d\tau$$ Here $K$ stand for the modified Bessel ...
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### Sum of Bessel function with integer parameters and fixed argument

Question. Let $J_{\nu}$ be a standart Bessel function of the first order. What is the asymptotic of the sum $\sum_{n\ge 0}|J_n(t)|$ as $t\to\infty$? An upper bounds stronger than $O(t)$ are also ...
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### Closed-form expression for definite integrals involving modified Bessel functions K_1 and K_0

I am attempting to derive a closed-form expression for the following two integrals involving the modified Bessel functions $K_1$ and $K_0$, but I can't find a solution (I don't know if there is one). ...
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### Matrix argument K Bessel functions at half integral orguments

As a working definition I will define: $$K_\nu^{(n)} (z) = \frac{1}{2^n}\int_{\mathcal{P}} e^{- \operatorname{tr}( y + y^{-1}) z/2} \det y^\nu d \mu(y)$$ where $\mathcal{P}$ represents the space of ...
• 21
1 vote
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### $n$th Derivative of $_p F_q(a_1,...,a_p; b_1,...,b_q;x^{-m})$, $p \le q$

Maple seems to suggest the following formula for $n>0$, $p \le q$: \begin{align} \frac{d^n}{d x^n} & {}_p F_q (a_1,\ldots,a_p;b_1,\ldots,b_q;1/x) \\[8pt] = {} & (-1)^n \hspace{1pt} n!\...
• 141
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### Bessel functions of matrix argument in the scalar case

Herz (1955) provides the following equality: $$A_\delta(-\lambda) - A_\delta(-\lambda)\lambda^{-\delta} = -\sin(\pi\delta)B_\delta(\lambda)/\pi$$ where $A_\delta$ and $B_\delta$ are the Bessel ...
• 2,353
1 vote
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### Approximation for a Bessel function integral

I'm trying to calculate hit probabilities on a dart board if the dart thrower has some Gaussian angle distribution function with width $\Delta$ and some systematic angle offsets $\phi_0, \theta_0$. ...
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### A bound for the Bessel function of the first kind J_0

I have proved the following bound for the Bessel function of the first kind: $$J_0(x)=\sum_{m=0}^\infty \frac{(-1)^m\,(x/2)^{2m}}{(m!)^2}$$ which is $$|J_0(x)|\le \frac1{\sqrt[4]{1+x^2}}$$ but I ...
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1 vote
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### Inverse mellin transform

Let $K_1(t)$ be the K-Bessel function, then we have $\int_{0}^{\infty}K_v(y)y^s\frac{dy}{y}=2^{s-2}\Gamma(\frac{s+v}{2})\Gamma(\frac{s-v}{2})$ See page 106 of Bump's book Automorphic forms and ...
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### L functions of Symmetric power of elliptic curves

Let $E$ be an elliptic curve over the raional field with conductor $N$, which corresponds to the eigenform $f(z)=\sum a_nq^n$. Let $L(Sym^2E,s)$ be the L function of the symmetric power of $E$.I am ...
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