ON A CLASS OF TWO-INDEX REAL HERMITE POLYNOMIALS ¨ JEDDA & ALLAL GHANMI NAIMA AIT
arXiv:1211.5745v1 [math.CA] 25 Nov 2012
A BSTRACT. We introduce a class of doubly indexed real Hermite polynomials and we deal with their related properties like the associated recurrence formulae, Runge’s addition formula, generating function and Nielsen’s identity.
1. I NTRODUCTION The Burchnall’s operational formula ([1]) m m d (−1)k Hm−k ( x ) dk ( f ), − + 2x ( f ) = m! ∑ dx k! (m − k )! dx k k =0
(1.1)
where Hm ( x ) denotes the usual Hermite polynomial ([2, 6]) m 2 d 2 Hm ( x ) = (−1)m e x e− x , (1.2) m dx enjoy a number of remarkable properties. It is used by Burchnall [1] to give a direct proof of Nielsen’s identity ([4]) min(m,n)
∑
Hm+n ( x ) = m!n!
k =0
(−2)k Hm−k ( x ) Hn−k ( x ) . k! (m − n)! (n − k)!
(1.3)
The special case of (1.1) where f = 1, i.e., Hm ( x ) =
d − + 2x dx
m
· (1).
(1.4)
can be employed to recover in a easier way the generating function +∞
∑
Hm ( x )
m =0
tm = exp(2xt − t2 ) m!
as well as the Runge addition formula ([5, 3]) √ √ m/2 n 1 Hk ( 2x ) Hm−k ( 2y) Hm ( x + y) = m! ∑ . 2 k! (m − k)! k =0
(1.5)
(1.6)
In this note, we have to consider the following class of doubly indexed real Hermite polynomials m d Hm,n ( x ) = − + 2x · ( x n ), (1.7) dx and we derive some of their useful properties. More essentially, we discuss the associated recurrence formulae, Runge’s addition formula, generating function and Nielsen’s identity. 2. D OUBLY INDEXED REAL H ERMITE POLYNOMIALS Hm,n ( x ) By taking f ( x ) = x n in (1.1), we obtain m d Hm,n ( x ) := − + 2x (xn ) dx min(m,n)
= m!n!
∑
k =0
(−1)k x n−k Hm−k ( x ) . k! (n − k)! (m − k)!
(2.1) (2.2)
Key words and phrases. Two-index Hermite polynomials; Runge’s addition formula; generating function; Nielsen’s identity. 1
¨ JEDDA & ALLAL GHANMI NAIMA AIT
2
It follows that Hm,n ( x ) is a polynomial of degree m + n, since Q( x ) := Hm,n ( x ) − x n Hm ( x ) is a polynomial of degree deg( Q) ≤ n + m − 2. For the unity of the formulations, we shall define trivially Hm,n ( x ) = 0 whenever m < 0 or n < 0. We call them doubly indexed real Hermite polynomials. Note that Hm,0 ( x ) = Hm ( x ), H0,n ( x ) = x n and ( 0 m n. Next, we state the following Proposition 2.6. The generating function of Hm,n is given by +∞ um vn ∑ Hm,n (x) m! n! = exp −u2 + (2u + v)x − uv . m,n=0 Proof. According to the definition of Hm,n , we can write " m # +∞ +∞ 1 um vn d · ∑ Hm,n (x) m! n! = ∑ m! −u dx + 2ux m =0 m,n=0 d = exp −u + 2ux (evx ) . dx
+∞
vn ∑ n! xn n =0
(2.14)
!
Making use of the Weyl identity which reads for the operators A = 2xId et B = −d/dx as exp(uA + uB) = exp(uA) exp(uB) exp −u2 Id ; u ∈ R, we get +∞
∑
Hm,n ( x )
m,n=0
2 um vn d = e2ux−u exp −u (evx ) . m! n! dx
Therefore, the desired result follows since ∞ d (−u)k d k vx vx (e ) = e−uv evx . exp −u (e ) = ∑ dx k! dx k =0
Remark 2.7. The special case of v = 0 (in (2.14)) infers the generating function (1.5) of the standard real Hermite polynomials Hm . Furthermore, for y = u = −v, we get e xy =
+∞
∑
(−1)n Hm,n ( x )
m,n=0
ym+n . m!n!
(2.15)
Proposition 2.8. We have the recurrence formula 0 Hm,n ( x ) = 2mHm−1,n ( x ) + nHm,n−1 ( x ).
(2.16)
Proof. Differentiating the both sides of (2.14) and making appropriate changes of indices yield (2.16). Corollary 2.9. We have ν Hr−ν+ j,n− j ( x ) dν , ( Hr,n ( x )) = r!n! ∑ α j,ν ν dx (r − ν + j ) ! ( n − j ) ! j =0
where α j,ν
ν 2 2α j,ν−1 + α j−1,ν−1 = 1
(2.17)
for j = 0 for 1 ≤ j < ν . for j = ν
Proof. This can be handled by mathematical induction using (2.16).
ON A CLASS OF TWO-INDEX REAL HERMITE POLYNOMIALS
5
Remark 2.10. The α j,ν are even positive numbers and their first values are j=0 1 2 22 23 24 25
α j,ν ν=0 ν=1 ν=2 ν=3 ν=4 ν=5
j=1
j=2
1 4 12 32 80
1 6 24 80
j=3
j=4
j=5
. 1 8 40
1 10
1
We conclude this note by giving a formula for the two-index Hermite polynomial Hm,n ( x ) expressing it as a weighted sum of a product of the same polynomials. Namely, we state the following Proposition 2.11. Keep notation as above. Then the Nielsen identity for Hm,n ; n ≥ 1, reads m,k,ν
Hm+r,n ( x ) = m!r!nn!
∑
α j,ν
k,ν,j=0
Γ(n + k − ν) (− x )ν Hm−k,n ( x ) Hr−ν+ j,n− j ( x ) . (k − ν)!ν! x n+k (m − k)!n! (r − ν + j)!(n − j)!
γ
Proof. Recall first that Hm ( x, α, p), the polynomials given through (2.8), can be rewritten in the following equivalent form ([7]) α m d γ γ −1 (1). + pγx − Hm ( x, α, p) := − dx x Now, since for the special values p = 1, γ = 2 and α = n, we have 2 Hm+r,n ( x ) = x n Hm +r ( x, n, 1) d = x n − + 2x − dx d n = x − + 2x − dx
n x
m
n x
m
Hr2 ( x, n, 1)
x −n Hr,n ( x ) ,
we can make use of the Burchnall’s formula extension proved by Gould and Hopper [7], to wit
−
d α + pγx γ−1 − dx x
m
γ (−1)k Hm−k ( x, α, p) dk ( f ). ∑ k! (m − k)! dx k k =0 m
( f ) = m!
Thus, for f = x −n Hr,n , we obtain m
Hm+r,n ( x ) = m!
(−1)k Hm−k,n ( x ) dk −n ( x Hr,n ( x )). k! (m − k)! dx k k =0
∑
(2.18)
Therefore, by applying the Leibnitz formula and appealing the result of Corollary 2.9, we get (−1)k Hm−k,n ( x ) k k dk−ν −n dν ∑ k! (m − k)! ∑ ν dxk−ν (x ) dxν ( Hr,n (x)) ν =0 k =0 m
Hm+r,n ( x ) = m!
m,k,ν
=m!r!nn!
∑
k,ν,j=0
α j,ν
Γ(n + k − ν) (− x )ν Hm−k,n ( x ) Hr−ν+ j,n− j ( x ) (k − ν)!ν! x n+k (m − k)!n! (r − ν + j)!(n − j)!
for every integer n ≥ 1. Note that for n = 0, (2.18) reads simply m
Hm+r ( x ) = m!
(−1)k Hm−k ( x ) dk ( Hr ( x )). k! (m − k )! dx k k =0
∑
In this case, we recover the usual Nielsen formula (1.3) for the real Hermite polynomials Hm .
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¨ JEDDA & ALLAL GHANMI NAIMA AIT
R EFERENCES [1] J. L. Burchnall, A note on the polynomials of Hermite. Quart. J. Math., Oxford Ser. 12, (1941). 9-11. [2] C. Hermite, Sur un nouveau dveloppement en srie des fonctions. Compt. Rend. Acad. Sci. Paris 58, p. 94-100 et 266-273, t. LVIII (1864) ou Oeuvres compltes, tome 2. Paris, p. 293-308, 1908. [3] J. Kamp de Friet, Sur une formule d’addition des polynomes d’Hermite. Volume 2 de Mathematisk-fysiske Meddelelser. 10 pages, Det Kgl. Danske Videnskabernes Selskab, Lunos, 1923 [4] N. Nielsen, Recherches sur les polynmes d’Hermite. Volume 1 de Mathematisk-fysiske meddelelser. 79 pages, Det Kgl. Danske Videnskabernes Selskab, 1918. [5] C. Runge, ber eine besondere Art von Intergralgleichungen, Math. Ann. 75 (1914) 130-132. [6] E.D. Rainville, Special functions. Chelsea Publishing Co., Bronx, N.Y., 1971. [7] H.W. Gould, A.T. Hopper, Operational formulas connected with two generalizations of Hermite polynomials. Duke Math. J. 29 1962 51-63. D EPARTMENT OF M ATHEMATICS , P.O. B OX 1014, FACULTY OF S CIENCES , M OHAMMED V-A GDAL U NIVERSITY, R ABAT, M OROCCO E-mail address:
[email protected]
