# An integral identity involving the Hermite polynomials.

[sections]1. Introduction

For any real number x, the polynomial solutions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

of the Hermite equations

[d.sup.2]y/[dx.sup.2] - 2x dy/dx + 2ny = 0 (n = 0,1,2, ...)

are called Hermite polynomials, see [1]. For example, the first several polynomials are: [H.sub.0](x) = 1, [H.sub.1](x) = 2x, [H.sub.2](x) = 4[x.sup.2] - 2, [H.sub.3](x) = 8[x.sup.3] - 12x, [H.sub.4](x) = 16[x.sup.4] - 48[x.sup.2] + 12, [H.sub.5](x) = 32[x.sup.5] - 160[x.sup.3] + 120x, .... It is well know that Hn(x) is an orthogonality polynomial. That is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

And it play a very important rule in the theories and applications of mathematics. So there are many people had studied its properties, some results and related papers see references [2], [3], [4], [5] and [6].

In this paper, we shall study the calculating problem of the integral

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and give an interesting calculating formula for it. About this problem, it seems that no one had studied yet, at least we have not seen any related papers before. The problem is interesting, because it can help us to know more information about the orthogonality of [H.sub.n] (x). The main purpose of this paper is using the elementary method and the properties of the power series to give an exact calculating formula for (1). That is, we shall prove the following conclusions:

Theorem 1. Let n and k are two positive integer with n [greater than or equal to] k, then we have the identity

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the summation over all nonnegative integers [a.sub.1], [a.sub.2], ... , [a.sub.k] such that [a.sub.1] + [a.sub.2] + ... [a.sub.k] = n.

Theorem 2. Let m, n and k are positive integers with n [greater than or equal to] k [greater than or equal to] 1, then we have the identity

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [H.sup.(m).sub.n](x) denotes the m-th derivative of [H.sub.n](x) for x.

From Theorem 1 and Theorem 2 we know that the integration must be 0, if n be an odd number. So it is interesting that the orthogonality in such an integral only depend on the parity of n.

[sections]2. Proof of the theorems

In this section, we shall use the elementary method and the properties of the power series to prove our Theorems directly. First we prove Theorem 1. For any positive integer k, from the generating function of [H.sub.n](x) and the properties of the power series we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

So from (2) we may get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

On the other hand, for any real number t and integer x, note that the integral

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Combining (2), (3), (4) and (5) we may get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

Comparing the coefficients of [t.sup.n] in (6) we may immediately deduce the identity

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the summation over all nonnegative integers [a.sub.1], [a.sub.2], ... , [a.sub.k] such that [a.sub.1] + [a.sub.2] + ... [a.sub.k] = n. This proves Theorem 1.

Now we prove Theorem 2. Note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then using the method of proving Theorem 1 we may get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [H.sup.(m).sub.n](x) denotes the m-th derivative of [H.sub.n](x) for x.

This completes the proof of Theorem 2.

References

[1] Guo Dajun etc., The Handbook of Mathematics, Shandong Science and Technology Press, Shandong, 1985.

[2] R. Willink, Normal moments and Hermite polynomials, Statistics & Probability Letters, 73(2005), 271-275.

[3] C. S. Withers, [lambda] simple expression for the multivariate Hermite polynomials, Statistics & Probability Letters, 47(2000), 165-169.

[4] Shi Wei, [lambda] globally uniform asymptotic expansion of the hermite polynomials, Acta Mathematica Scientia, 28(2008), 834-842.

[5] Hamza Chaggara, Operational rules and a generalized Hermite polynomials, Journal of Mathematical Analysis and Applications, 332(2007), 11-21.

[6] F. Brackx, H. De Schepper, N. De Schepper, F. Sommen, Two-index Clifford-Hermite polynomials with applications in wavelet analysis, Journal of Mathematical Analysis and Applications, 341(2008), 120-130.

[7] G. C. Greubel, On some generating functions for the extended generalized Hermite polynomials, Applied Mathematics and Computation, 173(2006), 547-553.

[8] Zhengyuan Wei, Xinsheng Zhang, Second order exponential differential operator and generalized Hermite polynomials, Applied Mathematics and Computation, 206(2008), 781-787.

Xiaoxia Yan

Hanzhong Vocational and Technical College, Hanzhong,

Shaanxi, 723000, P.R. China

(1) This work is supported by the N.S.F. (10671155) of P.R.China.