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It appears that associated Laguerre polynomials $L_n^{(\alpha)}(x^2)$ for half integer parameters in $n$ and $\alpha$ may be expressed in terms of an exponential function, an imaginary error function and polynomials of degree $2n$. For instance, with $\alpha=-1/2$ and various values of $n$:

  • $L_{1/2}^{(-1/2)}(x^2) = \frac{2}{\pi}e^{x^2} + \frac{2}{\sqrt{\pi}}(-x)\operatorname{erfi}(x)$
  • $L_{3/2}^{(-1/2)}(x^2) = \frac{4}{3\pi}(1-x^2)e^{x^2} + \frac{2}{3\sqrt{\pi}}(-3x+2x^3)\operatorname{erfi}(x)$
  • $L_{5/2}^{(-1/2)}(x^2) = \frac{4}{15\pi}(4-9x^2+2x^4)e^{x^2} + \frac{2}{15\sqrt{\pi}}(-15x+20x^3-4x^5)\operatorname{erfi}(x)$
  • $L_{7/2}^{(-1/2)}(x^2) = \frac{4}{105\pi}(24-87x^2+40x^4-4x^6)e^{x^2} + \frac{2}{105\sqrt{\pi}}(-105x+210x^3-84x^5+8x^7)\operatorname{erfi}(x)$

This seems to continue for arbitrary large $n$ with the same pattern. However, at functions.wolfram, the most extensive collection of formulas to my knowledge, I can't find any general relation for that. One can make a connection to confluent hypergeometric functions ${}_1F_1$, parabolic cylinder functions as well as Hermite polynomials, but still I cannot find an explicit expression for the stated formulas at arbitrary half-integer $n$.

Can anyone determine a formula for the polynomials in the above terms for arbitrary half-integer $n$?

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After some computations the final result is that $$ L_{n+1/2}^{(-1/2)}(-x) = p_n(x)\ \frac{e^{-x}}{ (2n+1)!!\ \pi/2} + q_n(x) \sqrt{x}\text{ erf}(\sqrt{x})\ \frac{2^{n+1} n!}{(2n+1)!\sqrt{\pi}} \tag{1} $$ where $\ p_n(x),\ q_n(x)\ $ are polynomials of degree $\ n\ $ with positive integer coefficients. $$ q_n(x) := 2^n\ n! \sum_{k=0}^n {n+1/2 \choose n-k}\ x^k/k!. \tag{2} $$ $p_n(x)\ $ is defined recursively by $$ p_0(x) = 1,\;\; p_1(x) = 2+2x, \\ p_n(x) = (4n-1 + 2x)\ p_{n-1}(x) + (2n-1)\ (2n-2)\ p_{n-2}(x). \tag{3} $$ The coefficients of $\ q_n(x)\ $ are OEIS sequence A223523. The coefficients of $\ p_n(x)\ $ are currently not in the OEIS.

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  • $\begingroup$ Thanks a lot! That's exactly what I was looking for. $\endgroup$ – bash4 Apr 15 at 9:48

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