# Associated Laguerre polynomials of half-integer parameters

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 $$n$$?

## 1 Answer

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.

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