Mediterr. J. Math. DOI 10.1007/s00009-015-0551-1 c Springer Basel 2015
A New Class of Generalized Polynomials Associated with Hermite and Euler Polynomials M. A. Pathan and Waseem A. Khan Abstract. In this paper, we introduce a new class of generalized polynomials associated with the modified Milne-Thomson’s polynomials (α) Φn (x, ν) of degree n and order α introduced by Dere and Simsek. The concepts of Euler numbers En , Euler polynomials En (x), generalized Euler numbers En (a, b), generalized Euler polynomials En (x; a, b, c) of Luo et al., Hermite–Bernoulli polynomials H E n (x, y) of Dattoli et al. (α) (α) and H E n (x, y) of Pathan are generalized to the one H E n (x, y, a, b, c) which is called the generalized polynomials depending on three positive real parameters. Numerous properties of these polynomials and some relationships between En , En (x), En (a, b), En (x; a, b, c) and (α) H En (x, y; a, b, c) are established. Some implicit summation formulae and general symmetry identities are derived using different analytical means and applying generating functions. Mathematics Subject Classification. 11B68, 33E20. Keywords. Hermite polynomials, Euler polynomials, Hermite–Euler polynomials, summation formulae, symmetric identities.
1. Introduction (α)
Dere and Simsek [4] modified the Milne-Thomson’s polynomials Φn (x) (see (α) for detail [9]) as Φn (x, ν) of degree n and order α by the means of the following generating function: g1 (t, x; α, ν) = f (t, α)ext+h(t,ν) =
∞ n=0
where f (t, α) is a function of t and integer α.
Φ(α) n (x, ν)
tn n!
(1.1)
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(α)
Observe that Φn (x, 0)= Φn (x) (cf. [9]). α in (1.1), we obtain the following polynomials Setting f (t, α) = et2+1 given by the generating function α ∞ (α) 2 En (x, ν)tn xt+h(t,ν) . (1.2) g2 (t, x; α, ν) = c = et + 1 n! n=0 (α)
Observe that the polynomials En (x, ν) are related to not only Bernoulli polynomials but also the Hermite polynomials. For example, if h(t, 0) = 0 in (1.2), we have En(α) (x, 0) = En(α) (x) (α)
where En (x, ν) denotes the Euler polynomials of higher order which is defined by means of the following generating function α ∞ 2 tn xt (α) FB (t, x; α) = . (1.3) e = E (x) n et + 1 n! n=0 One can easily see that En(α) (0, 0) = En(α) and, therefore,
FB (t; α) =
(α)
where En
2 t e +1
α =
∞ n=0
En(α)
tn n!
(1.4)
are generalized Euler numbers.
For more information about Euler numbers and Euler polynomials, we refer to [3,7,8,15,16]. If we take h(t, ν) = h(t, y) = yt2 in (1.1), we get gener(α) alized Hermite–Euler polynomials of two variables H En (x, y) (see Pathan [10] and Pathan et al. [11–13] for an analogous generalization of Hermite and Bernoulli polynomials) in the form α ∞ 2 tn xt+yt2 (α) (1.5) e = E (x, y) H n et + 1 n! n=0 which is essentially a generalization of Euler numbers, Euler polynomials, Hermite polynomials and Hermite–Euler polynomials H E (α) n (x, y) introduced by Dattoli et al. [3, p. 386 (1.6)] in the form ∞ 2 tn xt+yt2 . (1.6) = E (x, y) e n H et + 1 n! n=0 Luo et al. [7,8] generalized the concept of Euler numbers as follows Let a, b and a = b. The generalized Euler numbers En (a, b) for nonnegative integer n are defined by ∞ 2 tn (1.7) Φ(t; a, b) = t = En (a, b) , | t |< 2π. t a +b n! n=0
A New Class of Generalized Polynomials In [7] Luo et al. gave the following definition of the generalized Euler polynomials which generalize the concepts stated above. Let a, b, c > 0 and a = b. The generalized Euler polynomials En (x; a, b, c) for non-negative integer n are defined by Φ(x, t; a, b, c) =
∞ 2cxt tn = En (x; a, b, c) , t t a +b n! n=0
| t | < 2π.
(1.8)
It is easy to see that the above definition given by Luo et al. [8] is a natural and essential generalization of the concepts of Euler numbers En , Euler polynomials En (x) and the generalized Euler numbers En (a, b). For generating functions, related to non-negative real parameters a, b and c for the generalized Eulerian-type polynomials, we refer [15,16,18]. Some interesting generalizations of the families of Bernoulli, Euler and Genocchitype polynomials studied in the literature include [6–8,10–13,19]. Definition 1.1. Let c > 0. The generalized 2-variable 1-parameter Hermite Kamp’e de Feriet polynomials Hn (x, y, c) for non-negative integer n are defined by 2
cxt+yt =
∞
Hn (x, y, c)
n=0
tn . n!
(1.9)
This is an extended 2-variable Hermite Kamp’e de Feriet polynomials Hn (x, y) (see [1,2,9,17]) defined by 2
ext+yt =
∞
Hn (x, y)
n=0
tn . n!
(1.10)
Note that Hn (x, y, e) = Hn (x, y) To collect the powers of t we expand the left-hand side of (1.9) to get the representation [2] n n
Hn (x, y, c) =
j=0
j
(ln c)n−j xn−2j y j .
(1.11)
In this note, we first give definitions of the generalized Euler polynomi(α) als En (x, y; a, b.c) which generalize the concepts stated above and then research their basic properties and relationships with Euler numbers En , Euler polynomials En (x) and the generalized Euler numbers En (a, b), generalized Euler polynomials En (x; a, b, c) of Luo et al. [7] Hermite–Euler poly(α) nomials H En (x, y) of Dattoli et al. [3] and H En (x, y). The remainder of this paper is organized as follows. We modify generating functions for the Milne-Thomson’s polynomials [9] and derive some identities related to Hermite polynomials and Euler polynomials. Some implicit summation formulae and general symmetry identities are derived using different analytical means
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and applying generating functions. These results extend some known summations and identities of generalized Hermite, Bernoulli, Euler and Hermite– Euler polynomials studied by Dattoli et al. [3], Luo et al. [7,8], Yang et al. [20], Zhang et al. [21], Pathan [10] and Pathan et al. [11–13].
2. Definitions and Properties of the Generalized Euler Polynomials En(α) (x, y; a, b, c) In the modified Milne-Thomson’s polynomials αdue to Dere and Simsek 2 [4,9] defined by (1.1) if we set f (t, α) = at +bt , we obtain the following (α)
generalized polynomials En (x, ν; a, b, c). Definition 2.1. Let a, b, c > 0 and a = b. The generalized Euler polynomials (α) En (x, ν; a, b, c) for non-negative integer n are defined by α ∞ 2 tn xt+h(t,ν) (α) G1 (t, x; α, a, b, ν) = , c = E (x, ν; a, b, c) n at + bt n! n=0 | t |< 2π/(| ln a − ln b |),
x
(2.1)
2
Setting h(t, ν) = h(t, y) = yt in (2.1), we get Definition 2.2. Let a, b, c > 0 and a = b. The generalized Hermite–Euler (α) polynomials H En (x, y; a, b, c) for non-negative integer n are defined by α ∞ 2 tn xt+yt2 (α) G2 (t, x, y; α, a, b, c) = , c = H En (x, y; a, b, c) t t a +b n! n=0 | t |< 2π/(| ln a − ln b |),
x
For α = 1, we obtain from (2.2) the generating function ∞ 2 tn xt+yt2 G2 (t, x, y; 1, a, b, c) = t c = H En (x, y; a, b, c) t a +b n! n=0
(2.2)
(2.3)
whereas for x = 0 gives n
En(α) (0, y; a, b, c)
=
[2] k=0
n! (α) (ln c)k En−2k (a, b)y k . k!(n − 2k)!
(2.4)
Another special case of (2.2) for x = 0, y = 0 and c = e leads to the extension of the generalized Euler numbers En (a, b) for non-negative integer n defined by (1.7) in the form Definition 2.3. Let a, b > 0 and a = b. The generalized Euler polynomials (α) En (a, b) for non-negative integer n are defined by α 2 Φ(t; α, a, b) = at + bt ∞ tn = En(α) (a, b) , | t |< 2π/(| ln a − ln b |), x. n! n=0 (2.5)
A New Class of Generalized Polynomials It is easy to prove that Enα+β (a, b)
n n (β) (α) = (a, b)En−m (a, b). Em m
(2.6)
m=0
Further setting c = e in (2.2), we get Definition 2.4. Let a, b > 0 and a = b. The generalized Hermite–Euler poly(α) nomials H En (x, y; a, b, e) for non-negative integer n are defined by α ∞ 2 tn xt+yt2 (α) G2 (t, x, y; α, a, b, e) = , e = H En (x, y; a, b, e) t t a +b n! n=0 | t |< 2π/(| ln a − ln b |),
x .
(2.7)
(α)
The generalized Hermite–Euler polynomials H En (x, y; a, b, c) defined by (2.2) have the following properties which are stated as theorems below. Theorem 2.1. Let a, b, c > 0, a = b, xR and n ≥ 0. Then (α) H En (x, y, e, 1, e) En(α) (0, 0, e, 1, 1)
= H En(α) (x, y), En(α) (0, 0, a, b, 1) = En(α) (a, b),
= En(α) , En(1) (0, 0, a, b, 1) = En (a, b) (2.8) n n (α) (α+β) (β) E (x+y, z+u; a, b, c) = E (y, z; a, b, c)H Em (x, u; a, b, c) H n m H n−m m=0
(2.9) (α) H En (x + z, y; a, b, c) =
n n (α) En−m (z; a, b, c)Hm (x, y; c). (2.10) m
m=0
Proof. The formulas in (2.8) are obvious. Applying Definition 2.2, we have ∞ tn (α+β) (x + y, z + u; a, b, c) H En n! n=0 =
∞
t (α) H En (y, z; a, b, c)
n!
n=0 ∞
t (β) H Em (x, u; a, b, c)
m
m!
m=0
=
n
∞ ∞ n=0 m=0
t (α) (β) H En (y, z; a, b, c)H Em (x, u; a, b, c)
n+m
n!m!
.
Now replacing n by n − m and comparing the coefficients of tn , we get the result (2.9). Again by Definition 2.2, we have α ∞ 2 tn (x+z)t+yt2 (α) (2.11) c = E (x + z, y; a, b, c) H n at + bt n! n=0 which can be written as α ∞ ∞ 2 tn tm zt xt+yt2 (α) c c = E (z; a, b, c) Hm (x, y; a, b, c) . (2.12) n t t a +b n! m=0 m! n=0
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Replacing n by n − m in (2.12), comparing with (2.11) and equating their coefficients of tn leads to formula (2.10).
3. Implicit Summation Formulae Involving Generalized Euler and Generalized Hermite–Euler Polynomials (α)
For the derivation of implicit formulae involving generalized Euler En (α) (x, y; a, b, c) and generalized Hermite–Euler polynomials H En (x, y; a, b, e) the same considerations as developed for the ordinary Hermite and related polynomials in Khan et al. [5] and Hermite–Euler polynomials in Pathan [10] and Pathan et al. [11–13] holds as well. First we prove the following results (α) involving generalized Hermite–Euler polynomials H En (x, y; a, b, c). Theorem 3.1. Let a, b, c > 0 and a = b. Then for x, yR and n ≥ 0, the following implicit summation formulae for generalized Hermite–Euler polynomials (α) H En (x, y; a, b, c) holds true: (α) H Ek+l (z, y; a, b, c) =
k,l l k (α) (z − x)n+m H Ek+l−n−m (x, y; a, b, c). m n
n,m=0
(3.1) Proof. We replace t by t + u and rewrite the generating function (2.2) as α ∞ 2 tk u l (α) y(t+u)2 −x(t+u) . (3.2) c = c E (x, y; a, b, c) H k+l at+u + bt+u k! l! k,l=0
Replacing x by z in the above equation and equating the resulting equation to the above equation, we get c(z−x)(t+u)
∞
(α)
H Ek+l (x, y; a, b, c)
k,l=0
∞ tk u l = k! l!
ul . k! l!
t (α) H Ek+l (z, y; a, b, c)
k,l=0
k
(3.3) On expanding exponential function, (3.3) gives ∞ ∞ [(z − x)(t + u)]N N!
N =0
=
ul k! l!
t (α) H Ek+l (x, y; a, b, c)
k,l=0
∞
ul k! l!
t (α) H Ek+l (z, y; a, b, c)
k,l=0
k
k
(3.4)
which on using formula [17, p. 52 (2)] ∞ N =0
f (N )
∞ (x + y)N xn y m = f (n + m) N! n! m! n,m=0
(3.5)
A New Class of Generalized Polynomials in the left-hand side becomes ∞ ∞ (z − x)n+m tn um n!m! n,m=0
ul k! l!
t (α) H Ek+l (x, y; a, b, c)
k,l=0
∞
=
k
ul . k! l!
t (α) H Ek+l (z, y; a, b, c)
k,l=0
k
(3.6)
Now replacing k by k − n, l by l − m and using the lemma [17, p. 100 (1)] in the left-hand side of (3.6), we get ∞ ∞ ul (z − x)n+m tk (α) H Ek+l−n−m (x, y; a, b, c) n!m! (k − n)! (l − m)! n,p=0 k,l=0
=
∞
ul . k! l!
t (α) H Ek+l (z, y; a, b, c)
k,l=0
k
(3.7)
Finally on equating the coefficients of the like powers of t and u in the above equation, we get the required result. Remark 1. By taking l = 0 in Eq. (3.1), we immediately deduce the following result. Corollary 3.1. The following implicit summation formula for Hermite–Euler (α) polynomials H En (z, y; a, b, c) holds true: (α)
H Ek (z, y; a, b, c) =
k k n=0
n
(α)
(z − x)n H Ek−n (x, y, a, b, c).
(3.8)
Remark 2. On replacing z by z + x and setting y = 0 in Theorem 3.1, we get the following result involving generalized Euler polynomials of one variable (α)
Ek+l (z + x; a, b, c) =
k,l l k (α) (z)n+m Ek+l−m−n (x; a, b, c) m n
(3.9)
n,m=0
whereas by setting z = 0 in Theorem 3.1, we get another result involving generalized Euler polynomials of one and two variables (α)
Ek+l (y) =
k,l l k (α) (−x)n+m H Ek+l−m−n (x, y; a, b, c). (3.10) m n
n,m=0
Remark 3. Along with the above results we will exploit extended forms of (α) generalized Euler polynomials Ek+l (z; a, b, c) by setting y = 0 in the Theorem 3.1 to get (α)
Ek+l (z; a, b, c) =
k,l l k (α) (z − x)n+m Ek+l−m−n (x; a, b, c). (3.11) m n
n,m=0
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Remark 4. A straightforward expression of the H Ek+l (z, y; a, b, c) is suggested by a special case of the Theorem 3.1 for α = 1 in the following form k,l k l E (z, y; a, b, c) = (z − x)n+m H Ek+l−m−n (x, y; a, b, c) H k+l n m n,m=0
(3.12) where H Ek+l (z, y; a, b, c) denotes the generalized Euler polynomials defined by Luo et al. [7]. Theorem 3.2. Let a, b, c > 0 and a = b. Then for xR and n ≥ 0 n a b n (α) (α) (α−1) En (x + 1; a, b, c) = (a, b)En−m (x; , , c). Em m c c
(3.13)
m=0
Proof. We start with the definition α ∞ 2 tn (α) En (x+1; a, b, c) = t c(x+1)t t n! a + b n=0 α−1 2 2 = c(x+1)t at + bt at + bt and the result of Luo et al. [7, p. 3898 (4.16)] to get ∞
En(α) (x + 1; a, b, c)
n=0
∞ ∞ tn tn tm (α) a b (α−1) = Em (a, b) En (x; , , c) . n! m=0 m! n=0 c c n!
(3.14) n
Now replacing n by n − m and equating the coefficients of t leads to formula (3.13). Theorem 3.3. Let a, b, c > 0 and a = b. Then for x, yR and n ≥ 0 [n 2] a b n (α) j j (α) y (ln c) En−2j x; , , c . (3.15) H En (x + α, y; a, b, c) = 2j c c j=0
Proof. Since tn (α) = H En (x + α, y; a, b, c) n!
=
2 at + bt ∞
α
2
c(x+α)t+yt =
tn a b En(α) (x; , , c) c c n! n=0
2 a t ( c ) + ( cb )t
α 2
cxt cyt
⎞ ⎛ ∞ 2j t ⎠. ⎝ y j (ln c)j j! j=0
Now replacing n by n − 2j and comparing the coefficients of tn , we get the result (3.15). Remark. For α = 1, the above theorem reduces to [n 2] a b n (3.16) y j (ln c)j En−2j x; , , c H En (x + 1, y; a, b, c) = 2j c c j=0
A New Class of Generalized Polynomials whereas for y = 0, it reduces to the known result of Luo et al. [7, p. 3897 (4.12)] a b , , c . E (x + 1, y; a, b, c) = E x; H n n c c In a similar manner it is possible to find the explicit form of the generalized Bernoulli polynomials in terms of generalized Hermite polynomials which may yield generalizations of the results of Betti and Ricci [2] (see also [14,19]). Theorem 3.4. Let a, b, c > 0 and a = b. Then for x, yR and n ≥ 0 n n (α) (α) En−m (a, b)Hm (x, y, c). H En (x, y; a, b, c) = m
(3.17)
m=0
Proof. By the definition of generalized Euler polynomials and the definition (1.10), we have α ∞ 2 tn xt+yt2 (α) c = E (x, y; a, b, c) H n at + bt n! n=0 ∞ ∞ tn tm (α) = En (a, b) Hm (x, y; c) n! m! n=0 m=0 Now replacing n by n − m and comparing the coefficients of tn , we get the result (3.17). Remark. For c = e, (3.17) yields (α) H En (x, y; a, b, e) =
n n α (a, b)Hm (x, y). En−m m
m=0
Theorem 3.5. Let a, b, c > 0 and a = b. Then for x, yR and n ≥ 0 (α) H En (x, y; a, b, c)
=
[n n−2j 2]
(α)
(ln c)n−k−j Ek (a, b)
k=0 j=0
Proof. Applying the Definition 2.2 to the term 2
n! . (3.18) k!j!(n − 2j − k)!
2 at +bt
α and expanding the
exponential function cxt+yt at t = 0 yields α 2 2 cxt+yt t t a +b ⎞ ∞ ⎛ ∞ ∞ n 2j (α) t t tk ⎝ ⎠ = Ek (a, b) xn (ln c)n y j (ln c)j k! n! j! n=0 j=0 k=0 ⎛ ⎞ ∞ n ∞ n 2j t t n (α) ⎝ ⎠. = y j (ln c)j (ln c)n−k Ek (a, b)xn−k k n! j! n=0 j=0 k=0
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Replacing n by n − 2j in the r.h.s, we have ∞
t (α) H En (x, y; a, b, c)
n=0
=
∞ n=0
⎛ ⎝
n
n!
[n n−2j 2] k=0 j=0
n − 2j k
⎞ (α)
(ln c)n−k−j Ek (a, b)xn−k−2j y j ⎠
tn . (n − 2j)!j! (3.19) n
Combining (3.19) and (2.2) and equating their coefficients of t produce the formula (3.18). For y = 0, the above theorem reduces to the following result. Corollary 3.2. Let a, b, c > 0 and a = b. Then for x, yR and n ≥ 0 n n! (α) (α) En (x; a, b, c) = (ln c)n−k Ek (a, b)xn−k k!(n − k)! k=0
On the other hand if we set x = 0, the above theorem reduces to the result (2.4). Theorem 3.6. Let a, b, c > 0 and a = b. Then for x, yR and n ≥ 0 [n 2 ] n−2j n − 2j (α) (α) (ln c)n−k−j Ek (x; a, b, c). H En (x + 1, y; a, b, c) = k j=0 k=0
(3.20) Proof. By the definition of generalized Euler polynomials, we have α ∞ 2 tn (x+1)t+yt2 (α) (3.21) c = E (x + 1, y; a, b, c) H n at + bt n! n=0 ⎞ ∞ ∞ ⎛ ∞ k n 2j (α) t t ⎠ t ⎝ = Ek (x; a, b, c) xn (ln c)n y j (ln c)j k! n! j! n=0 j=0 k=0 ⎛ ⎞ n ∞ ∞ t2j ⎠ tn j n (α) = (3.22) y (ln c)j (ln c)n−k Ek (x; a, b, c) ⎝ k n! j=0 j! n=0 k=0
=
∞ ∞ n
n=0 j=0 k=0
n k
(α)
(ln c)n−k+j Ek (x; a, b, c)
tn+2j . n!j!
Replacing n by n − 2j, we have ∞ tn (α) H En (x + 1, y; a, b, c) n! n=0 [ 2 ] n−2j ∞ n − 2j n
=
n=0 j=0 k=0
k
(α)
(ln c)n−k−j Ek (x; a, b, c)
tn . (3.23) (n − 2j)!j!
Combining (3.21) and (3.23) and equating their coefficients of tn leads to formula (3.20).
A New Class of Generalized Polynomials Theorem 3.7. Let a, b > 0 and a = b. Then for x, yR and n ≥ 0 n n (α−1) (α) (α) (x, y; a, b, e). (3.24) En−m (a, b)H Em H En (x, y; a, b, e) = m m=0
Proof. Using the Definition 2.4 of the generalized Hermite–Euler polynomials, we can easily prove the formula (3.24) Theorem 3.8. For arbitrary real or complex parameter α, the following implicit summation formula involving generalized Euler polynomials (α) En (x, y; a, b, c) holds true: n n (α) (α) E (x + 1, y; a, b, c) = (3.25) (ln c)n−k H Ek (x, y; a, b, c). H n k k=0
Proof. By the definition of generalized Euler polynomials, we have ∞
∞ tn tn (α) + 1, y; a, b, c) − H En (x, y; a, b, c) n! n=0 n! n=0 α 2 2 cxt+yt (ct − 1) = at + bt ∞ ∞ ∞ n tk tn (α) nt (α) − E (x, y; a, b, c) (ln c) = H k H En (x, y; a, b, c) k! n! n! n=0 n=0 k=0 n ∞ ∞ n tn tn (α) (α) . = (ln c)n−k H Ek (x, y; a, b, c) − H En (x, y; a, b, c) k n! n=0 n! n=0 (α) H En (x
k=0
Finally, equating the coefficients of the like powers of tn , we get (3.25). Theorem 3.9. For arbitrary real or complex parameter α, the following implicit summation formula involving generalized Euler polynomials (α) En (x, y; a, b, c) holds true: n n (α) (ln ab)m (α)m H En−m (−x, y; a, b, c) = (−1)n H En(α) (x, y; a, b, c) m m=0 (3.26) (α) H En (x, y; a, b, c)
= (−1)n H En(α) (α − x, y; a, b, c).
(3.27)
Proof. We replace t by −t in (2.2) and then subtract the result from (2.2) itself finding
α 2 yt2 xt αt −xt (c − (ab) c ) c at + bt ∞ tn (3.28) = [1 − (−1)n ]H En(α) (x, y; a, b, c) n! n=0 which is equivalent to
M. A. Pathan and W. A. Khan ∞
tn (α) − H En (x, y; a, b, c) n! n=0 × =
∞
n
n
n!
n=0
n ∞
(α)m (ln ab)m
(α) H En−m (−x, y; a, b, c)
n=0 m=0
=
∞
n!
t (α) H En (x, y; a, b, c)
−
tm (α)m (ln ab)m m! m=0
t (α) H En (−x, y; a, b, c)
n=0 ∞
∞
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[1 − (−1)n ]H En(α) (x, y; a, b, c)
n=0
tn (n − m)!
tn n!
and thus by equating coefficients of like powers of tn , we get (3.26). To get (3.27), we write (3.28) in the form α αt 2 2 yt2 xt (α−x)t c − (c ) c at + bt ( cb )t + ( ac )t =
∞
[1 − (−1)n ]H En(α) (x, y; a, b, c)
n=0
tn n!
(3.29)
which is equivalent to ∞ ∞ tn c c tn (α) (α) − E (x, y; a, b, c) α − x, y; , , c H n H En n! n=0 b a n! n=0 =
∞
[1 − (−1)n ]H En(α) (x, y; a, b, c)
n=0
tn . n!
(3.30)
Now comparing the coefficients of tn in (3.30),we get (3.27). Remark 1. The formula n n (α) (α) (α)m H En−m (−x, y) = (−1)n H En−m (x, y) m
(3.31)
m=0
follows if in (3.26) we set a = c = e and b = 1. Remark 2. For a = c = e and b = 1, (3.27) reduces to the following result involving generalized Hermite–Euler polynomials (α) H En (α
(α)
− x, y) = (−1)n H En−m (x, y).
(3.32)
4. General Symmetry Identities In this section, we give general symmetry identities for the generalized (α) (α) Hermite–Euler polynomials H En (x, y; a, b, c) and Euler numbers En (a, b) by applying the generating functions (1.6) and (1.10). The results extend some known identities of Zhang et al. [21], Yang et al. [20] and Luo et al.
A New Class of Generalized Polynomials [7]. Throughout this section α will be taken as an arbitrary real or complex parameter. Theorem 4.1. Let a, b, c > 0 and a = b. Then for x, yR and n ≥ 0, the following identity holds true: n n
k
k=0
=
(α)
(α)
an−k bk H En−k (bx, b2 y; A, B, c)Ek (A, B)
n n
k
k=0
(α)
(α)
bn−k ak H En−k (ax, a2 y; A, B, c)Ek (A, B).
(4.1)
Proof. Start with g(t) =
22 (Aat + B at )(Abt + B bt )
α
2 2
cabxt+a
b yt2
.
(4.2)
Then the expression for g(t) is symmetric in a and b and we can expand g(t) into series in two ways to obtain g(t) = =
∞
(α) 2 H En (bx, b y; A, B, c)
n=0 n ∞
∞ (at)n (α) (bt)k Ek (A, B) n! k! k=0
(α) 2 H En−k (bx, b y; A, B, c)
n=0 k=0
an−k bk t n (α) Ek (A, B) . (n − k)! k!
On the similar lines we can show that g(t) = =
∞
∞ (bt)n (α) (at)k (α) 2 Ek (A, B) H En (ax, a x; A, B, c)
n=0 ∞ n
n!
(α) 2 H En−k (ax, a y; A, B, c)
n=0 k=0
k=0
k!
bn−k ak tn (α) Ek (A, B) . (n − k)! k!
By comparing the coefficients of tn on the right-hand sides of the last two equations we arrive the desired result. Remark 1. For α = 1, the above result reduces to n n k=0
=
k
an−k bk H En−k (bx, b2 y; A, B, c)Ek (A, B)
n n k=0
k
bn−k ak H En−k (ax, a2 y; A, B, c)Ek (A, B).
Further by taking c = e in Theorem 4.1, we immediately deduce the (α) following result involving generalized Hermite–Euler polynomials H En (x, y; a, b, e) for non-negative integer n
M. A. Pathan and W. A. Khan n n k=0
=
k
(α)
MJOM
(α)
an−k bk H En−k (bx, b2 y; A, B, e)Ek (A, B)
n n
k
k=0
(α)
(α)
bn−k ak H En−k (ax, a2 y; A, B, e)Ek (A, B)
(4.3)
Remark 2. By setting b = 1 in Theorem 4.1, we immediately get the following result n n (α) (α) an−k H En−k (x, y; A, B, c)Ek (A, B) k k=0 n n (α) (α) = (4.4) ak H En−k (ax, a2 y; A, B, c)Ek (A, B). k k=0
Theorem 4.2. Let a, b, c > 0 and a = b. Then for x, yR and n ≥ 0, the following identity holds true: b−1 n a−1 n (α) (−1)i+j an−k bk H En−k k i=0 j=0 k=0 b (α) × bx + i + j, b2 z; A, B, e Ek (ay; A, B, e) a n b−1 a−1 n (α) = (−1)i+j ak bn−k H En−k k i=0 j=0 k=0 a (α) × ax + i + j, a2 z; A, B, e Ek (by; A, B, e). (4.5) b Proof. Let
α 22 1 + (−1)a+1 eabt ab(x+y)t+a2 b2 zt2 e (Aat + B at )(Abt + B bt ) (eat + 1)(ebt + 1) α α 2 2 1 − (e−bt )a 1 − (e−at )b abxt+a2 b2 zt2 abyt g(t) = e e (Aat + B at ebt + 1 Abt + B bt eat + 1 g(t) =
=
(Aat
=
Aat
2 + B at
2 + B at
α
α
e
abxt+a2 b2 zt2
a−1
(−e
i=0
e
a2 b2 zt2
a−1 b−1
i
bti
)
Abt
2 + B bt
α
j (bx+ b i+j)at a
(−1) (−1) e
i=0 j=0
e
abyt
b−1
(−e
atj
)
(4.6)
j=0 ∞ k=0
(α)
Ek
(ay; A, B, e)
(bt)k k!
∞ a−1 ∞ b−1 (at)n (α) (bt)k b i+j (α) 2 (−1) Ek (ay; A, B, e) = bx + i + j, b z; A, B, e H En a n! (k)! n=0 i=0 j=0 k=0 =
a−1 b−1 ∞ n n (α) i j n−k k (−1) (−1) a b H En−k k n=0 k=0 i=0 j=0 b (α) 2 n bx + i + j, b z; A, B, e Ek (ay; A, B, e)t . a
Since (−1)a+1 = (−1)b+1 , the expression for α 22 1 + (−1)a+1 eabt ab(x+y)t+a2 b2 zt2 e g(t) = at at bt bt (A + B )(A + B ) (eat + 1)(ebt + 1)
(4.7)
A New Class of Generalized Polynomials is symmetric in a and b. Therefore, by symmetry we obtain the following power series expansion for g(t) g(t) =
∞ b−1 a−1 n n n=0 k=0
k
(α)
(−1)i+j bn−k ak H En−k
i=0 j=0
a (α) × ax + i + j, a2 z; A, B, e Ek (by; A, B, e)tn . b
(4.8)
By comparing the coefficients of tn on the right-hand sides of the last two equations, we arrive at the desired result. Acknowledgements The first author M. A. Pathan would like to thank Department of Science and Technology, Government of India, for the financial assistance for this work under Project Number SR/S4/MS:794/12.
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[email protected] Waseem A. Khan Department of Mathematics Integral University Lucknow 226026 India e-mail: waseem08
[email protected] Received: August 16, 2014. Revised: January 29, 2015. Accepted: February 16, 2015.
