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I am a physicist working on a problem where the following integrals are concerned:

$\int_{0}^{\infty} k^{l+1} e^{-p^2k^2}I_{\mu}(k)K_{l-\mu}(k)dk$

$\int_{0}^{\infty} k^{l+1} e^{-p^2k^2}(K_{\mu}(k))^2dk$

, such that $\mu, p \in \mathbb{R}$, $l\in\mathbb{N}^+$, $I_\mu(k)$ is the modified bessel function of the first kind and $K_\mu(k)$ is that of the second kind.

When $l=0$ this integral has a closed form as can be found here. I wonder for integers $l\geq 1$, which should be a straightforward generalization, if it also admits a closed form.

I am a physicist working on a problem where the following integrals are concerned:

$$\int_0^\infty k^{l+1} e^{-p^2k^2}I_\mu(k)K_{l-\mu}(k) \, dk$$

$$\int_0^\infty k^{l+1} e^{-p^2k^2}(K_\mu(k))^2 \, dk,$$

such that $\mu, p \in \mathbb{R}$, $l\in\mathbb{N}^+$, $I_\mu(k)$ is the modified bessel function of the first kind and $K_\mu(k)$ is that of the second kind.

When $l=0$ this integral has a closed form as can be found here. I wonder for integers $l\geq 1$, which should be a straightforward generalization, if it also admits a closed form.

I am a physicist working on a problem where the following integral is concerned:

$\int_{0}^{\infty} k^{l+1} e^{-p^2k^2}I_{\mu}(k)K_{l-\mu}(k)dk$

$\int_{0}^{\infty} k^{l+1} e^{-p^2k^2}(K_{\mu}(k))^2dk$

, such that $\mu, p \in \mathbb{R}$, $l\in\mathbb{N}^+$, $I_\mu(k)$ is the modified bessel function of the first kind and $K_\mu(k)$ is that of the second kind.

When $l=0$ this integral has a closed form as can be found here. I wonder for integers $l\geq 1$, which should be a straightforward generalization, if it also admits a closed form.

I am a physicist working on a problem where the following integrals are concerned:

$\int_{0}^{\infty} k^{l+1} e^{-p^2k^2}I_{\mu}(k)K_{l-\mu}(k)dk$

$\int_{0}^{\infty} k^{l+1} e^{-p^2k^2}(K_{\mu}(k))^2dk$

, such that $\mu, p \in \mathbb{R}$, $l\in\mathbb{N}^+$, $I_\mu(k)$ is the modified bessel function of the first kind and $K_\mu(k)$ is that of the second kind.

When $l=0$ this integral has a closed form as can be found here. I wonder for integers $l\geq 1$, which should be a straightforward generalization, if it also admits a closed form.

I am a physicist working on a problem where the following integral is concerned:

$\int_{0}^{\infty} k^{l+1} e^{-p^2k^2}I_{\mu}(k)I_{l-\mu}(k)dk$, such that $\mu, p \in \mathbb{R}$, $l\in\mathbb{N}^+$, and $I_\mu(k)$ is the modified bessel function of the first kind.

When $l=0$ this integral has a closed form as can be found here. I wonder for integers $l\geq 1$, which should be a straightforward generalization, if it also admits a closed form.

I am a physicist working on a problem where the following integral is concerned:

$\int_{0}^{\infty} k^{l+1} e^{-p^2k^2}I_{\mu}(k)K_{l-\mu}(k)dk$

$\int_{0}^{\infty} k^{l+1} e^{-p^2k^2}(K_{\mu}(k))^2dk$

, such that $\mu, p \in \mathbb{R}$, $l\in\mathbb{N}^+$, $I_\mu(k)$ is the modified bessel function of the first kind and $K_\mu(k)$ is that of the second kind.

When $l=0$ this integral has a closed form as can be found here. I wonder for integers $l\geq 1$, which should be a straightforward generalization, if it also admits a closed form.