I was searching in the Prudnikov (vol. 2) how to solve an integral and finally I found it. However, I didn't recognized a function that appears in the answer.

Integral 1.8.2.4:

$$ \int_0^x x^{\nu+1} e^{a x^2} J_{\nu}(b x) dx = \frac{b^\nu e^{a x^2}}{(2 i a)^{\nu+1}} \left[U_{\nu+1} (2iax^2, bx) + i U_{\nu+2} (2iax^2, bx)\right],~~~[Re{\nu} > -1] $$

I have searched what function is "$U_{\nu}(\cdot, \cdot)$" but I didn't found. Initially I believed that it was a Chebyshev polynomial, but it takes two parameters, not only one...

Anyone have an idea? I need to know in order to try to simplify the final expression.

I was searching in the Prudnikov (vol. 2) how to solve an integral and I finally found it. However, I didn't recognized a function that appears in the answer.

Integral 1.8.2.4:

$$ \int_0^x x^{\nu+1} e^{a x^2} J_{\nu}(b x) dx = \frac{b^\nu e^{a x^2}}{(2 i a)^{\nu+1}} \left[U_{\nu+1} (2iax^2, bx) + i U_{\nu+2} (2iax^2, bx)\right],~~~[Re{\nu} > -1] $$

I have searched what function is "$U_{\nu}(\cdot, \cdot)$" but I didn't found. Initially I believed that it was a Chebyshev polynomial, but it takes two parameters, not only one...

Anyone have an idea? I need to know in order to try to simplify the final expression.