Some recurrence formulas for the Hermite polynomials and their squares

In this paper, by making use of the generating function methods and Padé approximation techniques, we establish some new recurrence formulas for the Hermite polynomials and their squares. These results presented here are the corresponding extensions of some known formulas.


Introduction
In the Sturm-Liouville boundary value problem, there is a special case called Hermite's di erential equation which arises in the treatment of the harmonic oscillator in quantum mechanics. It is well known that Hermite's di erential equation is de ned as where n is a real number. In particular, for non-negative integer n, the solutions of Hermite's di erential equation are usually referred to as the Hermite polynomials H n (x), which are de ned by means of the exponential generating function It is easily seen from (2) that the Hermite polynomials can be determined by H n (x) = ∂ n ∂t n exp( xt − t ) t= (n ≥ ).
The rst several Hermite polynomials are These polynomials have played important roles in various elds of mathematics, physics and engineering, such as quantum mechanics, mathematical physics, ucleon physics and quantum optics. It is clear that the Poisson kernel for the Hermite polynomials is (see, e.g., [8]) In particular, the case x = y in (5) yields the squares H n (x) of the Hermite polynomials given by It is easily seen that (6) can be reformulated as Recently, Kim et al. [9][10][11][12], Qi and Guo [17] studied the generating functions of the Hermite polynomials and their squares, and presented some explicit formulas for the Hermite polynomials and their squares. Further, Qi and Guo [17] used the properties of the Bell polynomials of the second kind stated in [16] to obtain some explicit formulas and recurrence relations for the Hermite polynomials and their squares, for example, they showed that for non-negative integer n, the Hermite polynomials and their squares can be computed by and and there exist the following recurrence formulas for the Hermite polynomials and their squares, as follows, and where, and in what follows, a k is the binomial coe cient de ned for complex number a and non-negative Motivated and inspired by the work of Kim et al. [9], Qi and Guo [17], in this paper we establish some new recurrence formulas for the Hermite polynomials and their squares by making use of the generating function methods and Padé approximation techniques. It turns out that the formulas (8), (9) and (11) and an analogous formula to (10) described in [9] are derived as special cases.

Padé approximants
We here recall the de nition of Padé approximation to general series and their expression in the case of the exponential function, which have been widely used in various elds of mathematics, physics and engineering; see, for example [3,13]. Let m, n be non-negative integers and let P k be the set of all polynomials of degree ≤ k. Assume that f is a function given by a Taylor expansion (13) in a neighborhood of the origin, a Padé form of type (m, n) is the following pair (P, Q) such that and It is clear that every Padé form of type (m, n) for f (t) always exists and satis es the same rational function, and the uniquely determined rational function P Q is usually called the Padé approximant of type (m, n) for f (t) (see, e.g., [1,4]). For non-negative integers m, n, the Padé approximant of type (m, n) for the exponential function exp(t) is the unique rational function (see, e.g., [7,14]) which obeys the property In fact, the explicit formulas for P m and Q n can be expressed in the following way (see, e.g., [2,15]): and where P m (t) and Q n (t) is called the Padé numerator and denominator of type (m, n) for the exponential function exp(t), respectively. We shall use the above properties of Padé approximants to the exponential function to establish some new recurrence formulas for the Hermite polynomials and their squares in next section.

Theorem 3.1. Let m, n be non-negative integers. Then, for non-negative integer l with
Proof. Let m, n be non-negative integers. If we denote the right hand side of (20) by S m,n (t) then we have It is easily seen from (2) that By applying (22) to (23), we discover which can be rewritten as We now apply the exponential series exp(xt) = ∑ ∞ k= x k t k k! in the right hand side of (20). With the help of the beta function, we get For convenience, let p m,n;k , q m,n;k and s m,n;k be the coe cients of the polynomials P m (t), Q n (t) and S m,n (t) given by P m (t) = m k= p m,n;k t k , Q n (t) = n k= q m,n;k t k , and S m,n (t) = ∞ k= s m,n;k t m+n+k+ .
It follows from (18), (19) and (26) that and If we apply (27) and (28) which together with the Cauchy product yields By comparing the coe cients of t l in (32), we obtain that for non-negative integer l with ≤ l ≤ (m + n) + , Thus, applying (29) to (33) gives the desired result.
We next discuss some special cases of Theorem 3.1. By taking m = in Theorem 3.1, we obtain that for nonnegative integer l with ≤ l ≤ n + , which means If we take n = in Theorem 3.1, we obtain that for non-negative integer l with ≤ l ≤ m + , which implies (8), and (37) can be regarded as an analogous version of the formula (10). In fact, (35) and (37) were rediscovered by Kim et al. [9] where some interesting identities between the Hermite polynomials and the Bernoulli and Euler polynomials can be also found. We here refer to [5] for some analogous formulas for the generalized Hermite polynomials to (35) and (37).

Theorem 3.3.
Let m, n be non-negative integers. Then, for non-negative integer l with l ≥ (m + n + ), Proof. It is easily seen that comparing the coe cients of t l in (32) gives that for l ≥ (m + n + ), Thus, by applying (29) and (30) to (39), we obtain the desired result.
In particular, the case l = (m + n + ) in Theorem 3.3 gives that for non-negative integers m, n, If we take m = in (40), we get that for non-negative integer n, n! ( n + )! H n+ (x) + (− ) n n! (n + )! = i+k= n+ i,k≥ It is clear that (41) is the case l = n + in (35).