SOME PROPERTIES OF THE HERMITE POLYNOMIALS AND THEIR SQUARES AND GENERATING FUNCTIONS

In the paper, the authors consider the generating functions of the Hermite polynomials and their squares, present explicit formulas for higher order derivatives of the generating functions of the Hermite polynomials and their squares, which can be viewed as ordinary differential equations or derivative polynomials, find differential equations that the generating functions of the Hermite polynomials and their squares satisfy, and derive explicit formulas and recurrence relations for the Hermite polynomials and their squares.

In [3, p. 250], it was given that the squares H 2 n (x) for n ≥ 0 of the Hermite polynomials H n (x) can be generated by 1 In [5], the equation ( 2) was reformulated as Indeed, this is a typo and the corrected one should be After inductively arguing for nine pages, it was obtained in [5, Theorem 1] that the ordinary differential equations for n ≥ 0 have the same solution where and From [5, Theorem 1] mentioned above, Theorems 2 and 3 in [5], which can be corrected as j=n−i p+q+r=k (−1) q k p, q, r (i + p − 1) p (j + q − 1) q a i,j (n, x) for k, n ≥ 0, were derived, where denotes the rising factorial and It is clear that the quantities a i,j (n, x) in [5] were expressed by a recurrent relation and can not be computed easily by hand and by computer softwares.We observe that, when k = 2n − j − 2i and i + j = n, the quantity a i,j−k−2 (n − k − 1, x) in the recurrence relation (5) becomes which implies that Theorem 1, consequently Theorems 2 and 3, in [5], are wrong.
In this paper, we will reconsider the generating functions e 2tx−t 2 and F (t) = F (t, x) defined in (4), present explicit formulas for the nth derivatives of the functions F (t) and e 2tx−t 2 , which can be viewed as ordinary differential equations or derivative polynomials [7], find more differential equations that the functions F (t) and e 2tx−t 2 satisfy, and derive explicit formulas and recurrence relations for the Hermite polynomials H n (x) and their squares H 2 n (x).The main results of this paper can be stated as the following theorems.
Theorem 1.1.For n ≥ 0, the nth derivative of the function F (t) = F (t, x) defined in (4) can be computed by where 0 0 = 1 and p q = 0 for q > p ≥ 0. Theorem 1.2.For n ≥ 0, the squares H 2 n (x) of the Hermite polynomials H n (x) can be computed by Theorem 1.3.For n ≥ 0, the Hermite polynomials H n (x) can be computed by and the nth derivative of their generating function e 2xt−t 2 can be computed by Theorem 1.4.For n ≥ 0, the Hermite polynomials H n (x) and their derivatives for n ≥ 2.

Preprints
and For n ≥ 0, the squares H 2 n (x) of the Hermite polynomials H n (x) satisfy the recurrence relations and

Lemmas
In order to prove our main results, we need several lemmas below.
The Faà di Bruno formula can be described in terms of the Bell polynomials of the second kind B n,k (x 1 , x 2 , . . ., x n−k+1 ) by Lemma 2.2 ([2, p. 135]).For complex numbers a and b, we have , and [10, Section 3]).For 0 ≤ k ≤ n, the Bell polynomials of the second kind B n,k satisfy Lemma 2.4 ([2, p. 135, Theorem B] and [6, Theorem 1.1]).For n ≥ k ≥ 0, we have and for j ≥ k ≥ 0. Equivalently and unifiedly, where for arbitrary a ∈ C and k ≥ 0 and α k is called the falling factorial.
Proof.In [2, p. 133], it was listed that for k ≥ 0. From this, it follows that .
Further differentiating m ≥ k times and making use of ( 16), (17), and (18) yield which is equivalent to (20) and which is not simpler than the nice expression (23).

Proofs of main results
Now we are in a position to prove our main results.
The formula ( 6) is thus proved.The proof of Theorem 1.1 is complete.
Proof of Theorem 1.2.Since − 1, employing the L'Hôspital rule and the formula (23) leads to lim Therefore, taking the limit t → on both sides of (6) yields which means by (3) that This can be rearranged as (7).The proof of Theorem 1.2 is complete.
where u = u(t) = 2xt − t 2 .Hence, we acquire The formula (8) follows.This formula can also be derived similarly by considering The proof of Theorem 1.3 is complete.
Proof of Theorem 1.4.Differentiating with respect to x on both sides of (1) yields Hence, it follows that H 0 (x) = 0 and the formula (9) is valid.Differentiating with respect to x on both sides of (2) gives This means that H 0 (x) = 0 and for n ∈ N, which can be simplified as Combining this with (9) derives the formula (10).
Proof of Theorem 1.5.It is easy to see that e Further taking the limit t → 0 yields .
Further taking the limit t → 0 results in