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Journal of Integer Sequences, Vol. 17 (2014),
Article 14.1.2
23 11
A New Approach to Multivariate q-Euler
Polynomials Using the Umbral Calculus
Serkan Araci
Atatürk Street
31290 Hatay
Turkey
mtsrkn@hotmail.com
Xiangxing Kong
Department of Mathematics and Statistics
Central South University
Changsha 410075
China
xiangxingkong@gmail.com
Mehmet Acikgoz
Department of Mathematics
University of Gaziantep
27310 Gaziantep
Turkey
acikgoz@gantep.edu.tr
Erdoğan Şen
Department of Mathematics
Namik Kemal University
59030 Tekirdağ
Turkey
erdogan.math@gmail.com
Abstract
We derive numerous identities for multivariate q-Euler polynomials by using the
umbral calculus.
1
1
Preliminaries
Throughout this paper, we use the following notation, where C denotes the set of complex numbers,
F denotes the seto of all formal power series in the variable t over C with
n
P∞
k
F = f (t) = k=0 ak tk! | ak ∈ C , P = C[x] and P ∗ denotes the vector space of all linear
functional on P, hL | p(x)i denotes the action of the linear functional L on the polynomial
p(x), and it is well-known that the vector space operation on P ∗ is defined by
hL + M | p(x)i = hL | p(x)i + hM | p(x)i ,
hcL | p(x)i = c hL | p(x)i ,
where c is some constant in C (for details, see [5, 6, 8, 11]).
The formal power series are known by the rule
f (t) =
∞
X
k=0
ak
tk
∈F
k!
which defines a linear functional on P as hf (t) | xn i = an for all n ≥ 0 (for details, see
[5, 6, 8, 11]]). Additionally,
tk | xn = n!δn,k ,
(1)
P∞
k
where δn,k is the Kronecker symbol. When we take fL (t) = k=0 L | xk tk! , then we obtain
hfL (t) | xn i = hL | xn i and so as linear functionals L = fL (t) (see [5, 6, 8, 11]). Additionally,
the map L → fL (t) is a vector space isomorphism from P ∗ onto F. Henceforth, F denotes
both the algebra of the formal power series in t and the vector space of all linear functionals
on P, and so an element f (t) of F can be thought of as both a formal power series and a
linear functional. The algebra F is called the umbral algebra (see [5, 6, 8, 11]).
Also, the evaluation functional for y in C is defined to be power series eyt . We can write
that heyt | xn i = y n and so heyt | p(x)i = p(y) (see [5, 6, 8, 11]). We note that for all f (t) in
F
∞
X
tk
f (t) =
f (t) | xk
(2)
k!
k=0
and for all polynomials p(x),
p(x) =
∞
X
tk | p(x)
k=0
xk
,
k!
(3)
(for details, see [5, 6, 8, 11]). The order o(f (t)) of the power series f (t) 6= 0 is the smallest
integer k for which ak does not vanish. It is considered o(f (t)) = ∞ if f (t) = 0. We see that
o(f (t)g(t)) = o(f (t)) + o(g(t)) and o(f (t) + g(t)) ≥ min {o(f (t)), o(g(t))}. The series f (t)
1
has a multiplicative inverse, denoted by f (t)−1 or f (t)
, if and only if o(f (t)) = 0. Such series
2
is called an invertible series. A series f (t) for which o(f (t)) = 1 is called a delta series (see
[5, 6, 8, 11]). For f (t), g(t) ∈ F, we have hf (t)g(t) | p(x)i = hf (t) | g(t)p(x)i.
A delta series f (t) has a compositional inverse f (t) such that f f (t) = f (f (t)) = t.
For f (t), g(t) ∈ F , we have hf (t)g(t) | p(x)i = hf (t) | g(t)p(x)i. By (3), we have
∞
p(k) (x) =
dk p(x) X tl | p(x)
l (l − 1) · · · (l − k + 1) xl−k .
=
dxk
l!
l=k
(4)
Thus, we see that
p(k) (0) = tk | p(x) = 1 | p(k) (x) .
(5)
By (4), we get
tk p(x) = p(k) (x) =
dk p(x)
.
dxk
(6)
So, we have
eyt p(x) = p (x + y) .
(7)
Let f (t) be a delta series and let g(t) be an invertible series. Then there exists a unique
sequence Sn (x) of polynomials, with deg Sn (x) = n, such that g(t)f (t)k | Sn (x) = n!δn,k
for all n, k ≥ 0. The sequence Sn (x) is called the Sheffer sequence for (g(t), f (t)) or that
Sn (t) is Sheffer for (g(t), f (t)).
The Sheffer sequence for (1, f (t)) is called the associated sequence for f (t); we also say
Sn (x) is associated with f (t). The Sheffer sequence for (g(t), t) is called the Appell sequence
for g(t); we also say Sn (x) is Appell for g(t).
Let p(x) ∈ P. Then we have
yt
Z y
e −1
p(u)du,
| p(x)
=
t
0
hf (t) | xp(x)i = h∂t f (t) | p(x)i = hf´(t) | p(x)i ,
(8)
eyt − 1 | p(x) = p(y) − p(0), (see [5, 6, 8, 11]).
Let Sn (x) be Sheffer for (g (t) , f (t)). Then the following results are known in [11]:
h (t) =
∞
X
hh (t) | Sk (x)i
k=0
k!
g (t) f (t)k , h (t) ∈ F
∞
X
g (t) f (t)k | p(x)
Sk (x) , p(x) ∈ P,
p(x) =
k!
k=0
∞
X
1
tk
yf (t)
e
=
Sk (y) , for all y ∈ C,
k!
g f (t)
k=0
f (t)Sn (x) = nSn−1 (x).
3
(9)
Let a1 , . . . , ar , b1 , . . . , br be positive integers. Kim and Rim [1] defined the generating
function for multivariate q-Euler polynomials as follows:
Fq (t, x | a1 , . . . , ar ; b1 , . . . , br ) =
∞
X
En,q (x | a1 , . . . , ar ; b1 , . . . , br )
n=0
r
=
(q b1 ea1 t
tn
n!
(10)
2
ext .
a
t
b
r
r
+ 1) · · · (q e + 1)
Note that
E0,q (x | a1 , . . . , ar ; b1 , . . . , br ) =
2r
[2]qb1 [2]qb2 · · · [2]qbr
,
where [x]q is q-extension of x defined by
[x]q =
qx − 1
= 1 + q + q 2 + · · · + q x−1 .
q−1
We assume that q ∈ C with |q| < 1. Also, we note that limq→1 [x]q = x (see [1]–[11]). In
the special case, x = 0, En,q (0 | a1 , . . . , ar ; b1 , . . . , br ) := En,q (a1 , . . . , ar ; b1 , . . . , br ) are called
multivariate q-Euler numbers. By (10), we obtain the following:
En,q (x | a1 , . . . , ar ; b1 , . . . , br ) =
n X
n
k
k=0
xk En−k,q (a1 , . . . , ar ; b1 , . . . , br ) .
(11)
Kim and Kim [5] studied some interesting identities for Frobenius-Euler polynomials
arising from umbral calculus. They derived not only new but also fascinating identities in
modern classical umbral calculus.
By the same motivation, we also get numerous identities for multivariate q-Euler polynomials by utilizing from the umbral calculus.
2
On the multivariate q-Euler polynomials arising from
umbral calculus
Assume that Sn (x) is an Appell sequence for g (t). By (9), we have
1 n
x = Sn (x) if and only if xn = g(t)Sn (x), (n ≥ 0) .
g(t)
Let us take
q b 1 e a1 t + 1 · · · q b r e ar t + 1
g (t | a1 , . . . , ar ; b1 , . . . , br ) =
∈ F.
2r
4
(12)
Then we readily see that g (t | a1 , . . . , ar ; b1 , . . . , br ) is an invertible series. By (12), we
have
∞
X
tn
1
En,q (x | a1 , . . . , ar ; b1 , . . . , br )
=
ext .
(13)
n!
g
(t
|
a
,
.
.
.
,
a
;
b
,
.
.
.
,
b
)
1
r
1
r
n=0
By (13), we obtain the following
1
xn = En,q (x | a1 , . . . , ar ; b1 , . . . , br ) .
g (t | a1 , . . . , ar ; b1 , . . . , br )
(14)
Also, by (6), we have
tEn,q (x | a1 , . . . , ar ; b1 , . . . , br ) = E´n,q (x | a1 , . . . , ar ; b1 , . . . , br )
= nEn−1,q (x | a1 , . . . , ar ; b1 , . . . , br ) .
(15)
By (14) and (15), we have the following proposition.
Proposition 1. For n ≥ 0, En,q (x | a1 , . . . , ar ; b1 , . . . , br ) is an Appell sequence for
q b 1 e a1 t + 1 · · · q b r e ar t + 1
.
g (t | a1 , . . . , ar ; b1 , . . . , br ) =
2r
By (10), we see that
∞
X
En+1,q (x | a1 , . . . , ar ; b1 , . . . , br )
n=1
tn
xgext − g´ext
=
n!
g2
∞ X
1 n g´1 n tn
=
x
x x −
g
gg
n!
n=0
(16)
where we used g := g (t | a1 , . . . , ar ; b1 , . . . , br ). Because of (14) and (16), we discover the
following:
En+1,q (x | a1 , . . . , ar ; b1 , . . . , br )
g´
= xEn,q (x | a1 , . . . , ar ; b1 , . . . , br ) − En,q (x | a1 , . . . , ar ; b1 , . . . , br ) .
g
(17)
Therefore, we deduce the following theorem.
(qb1 ea1 t +1)···(qbr ear t +1)
∈ F. Then we have
Theorem 2. Let g := g (t | a1 , . . . , ar ; b1 , . . . , br ) =
2r
for n ≥ 0 :
g´
En+1,q (x | a1 , . . . , ar ; b1 , . . . , br ) = x −
En,q (x | a1 , . . . , ar ; b1 , . . . , br ) .
(18)
g
5
From (10), we derive that
∞
X
n=0
tn
q En,q (x + ar | a1 , . . . , ar ; b1 , . . . , br ) + En,q (x | a1 , . . . , ar ; b1 , . . . , br )
n!
br
=2
∞
X
En,q (x | a1 , . . . , ar−1 ; b1 , . . . , br−1 )
n=0
By comparing the coefficients in the both sides of
tn
n!
(19)
tn
.
n!
on the above, we obtain the following
2En,q (x | a1 , . . . , ar−1 ; b1 , . . . , br−1 ) = q br En,q (x + ar | a1 , . . . , ar ; b1 , . . . , br )
+En,q (x | a1 , . . . , ar ; b1 , . . . , br ) .
(20)
From Theorem 2, we get the following equation
gEn+1,q (x | a1 , . . . , ar ; b1 , . . . , br )
= gxEn,q (x | a1 , . . . , ar ; b1 , . . . , br ) − g´En,q (x | a1 , . . . , ar ; b1 , . . . , br ) .
(21)
By using (20) and (21), we arrive at the desired theorem.
Theorem 3. For n ≥ 0, we have
2En,q (x | a1 , . . . , ar−1 ; b1 , . . . , br−1 ) = q br En,q (x + ar | a1 , . . . , ar ; b1 , . . . , br )
+En,q (x | a1 , . . . , ar ; b1 , . . . , br ) .
Now, we consider that
Z x+y
En,q (u | a1 , . . . , ar ; b1 , . . . , br ) du
x
1
(En+1,q (x + y | a1 , . . . , ar ; b1 , . . . , br ) − En+1,q (x | a1 , . . . , ar ; b1 , . . . , br ))
n+1
∞ 1 X n+1
En+1−j,q (x | a1 , . . . , ar ; b1 , . . . , br ) y j
=
n + 1 j=1
j
=
=
∞
X
n (n − 1) (n − 2) · · · (n − j + 2)
j!
j=1
1
=
t
yt
=
∞
X
y j tj
j=0
j!
En+1−j,q (x | a1 , . . . , ar ; b1 , . . . , br ) y j
!
− 1 En,q (x | a1 , . . . , ar ; b1 , . . . , br )
e −1
En,q (x | a1 , . . . , ar ; b1 , . . . , br ) .
t
Therefore, we discover the following theorem:
6
(22)
Theorem 4. For n ≥ 0, we have
Z x+y
eyt − 1
En,q (x | a1 , . . . , ar ; b1 , . . . , br ) .
En,q (u | a1 , . . . , ar ; b1 , . . . , br ) du =
t
x
By (15) and Proposition 1, we have
1
En+1,q (x | a1 , . . . , ar ; b1 , . . . , br ) = En,q (x | a1 , . . . , ar ; b1 , . . . , br ) .
t
n+1
Thanks to (24), we readily derive the following:
En+1,q (x | a1 , . . . , ar ; b1 , . . . , br )
yt
e −1|
n+1
yt
e −1
En+1,q (x | a1 , . . . , ar ; b1 , . . . , br )
=
|t
t
n+1
yt
e −1
=
| En,q (x | a1 , . . . , ar ; b1 , . . . , br ) .
t
(23)
(24)
(25)
On account of (8) and (24), we get
yt
En+1,q (x | a1 , . . . , ar ; b1 , . . . , br )
e −1
yt
| En,q (x | a1 , . . . , ar ; b1 , . . . , br ) = e − 1 |
t
n+1
1
{En+1,q (y | a1 , . . . , ar ; b1 , . . . , br ) − En+1,q (a1 , . . . , ar ; b1 , . . . , br )}
=
n+1
Z
y
En,q (u | a1 , . . . , ar ; b1 , . . . , br ) du.
=
0
Consequently, we obtain the following theorem.
Theorem 5. For n ≥ 0, we have
yt
Z y
e −1
En,q (u | a1 , . . . , ar ; b1 , . . . , br ) du.
| En,q (x | a1 , . . . , ar ; b1 , . . . , br ) =
t
0
(26)
Assume that
P (q | a1 , . . . , ar ; b1 , . . . , br ) = {p(x) ∈ Q (q | a1 , . . . , ar ; b1 , . . . , br ) [x] | deg p(x) ≤ n}
is a vector space over Q (q | a1 , . . . , ar ; b1 , . . . , br ) which are the space of all polynomials
including coefficients q, a1 , . . . , ar , b1 , . . . , br .
For p(x) ∈ P (q | a1 , . . . , ar ; b1 , . . . , br ), let us consider
p(x) =
n
X
bk Ek,q (x | a1 , . . . , ar ; b1 , . . . , br ) .
k=0
7
(27)
By Proposition 1, En,q (u | a1 , . . . , ar ; b1 , . . . , br ) is an Appell sequence for
q b 1 e a1 t + 1 · · · q b r e ar t + 1
.
g := g (t | a1 , . . . , ar ; b1 , . . . , br ) =
2r
Thus we have
g (t | a1 , . . . , ar ; b1 , . . . , br ) tk | En,q (x | a1 , . . . , ar ; b1 , . . . , br ) = n!δn,k .
(28)
From (27) and (28), we compute
k
g (t | a1 , . . . , ar ; b1 , . . . , br ) t | p(x) =
n
X
bl gtk | El,q (x | a1 , . . . , ar ; b1 , . . . , br )
(29)
l=0
=
n
X
bl l!δl,k = k!bk .
l=0
Thus, by (29), we derive
1
gtk | p(x)
k!
1
q b1 ea1 t + 1 · · · q br ear t + 1 | p(k) (x) .
= r
2 k!
It is not difficult to show the following
X
Pr
Pr
q b 1 e a1 t + 1 · · · q b r e ar t + 1 =
q l=1 bl kl et j=1 aj kj .
bk =
k1 ,...,kr ≥0
k1 +k2 +...+kr =1
Via the results (30) and (31), we easily see that
D Pr
E
X
Pr
1
b l kl
t j=1 aj kj
(k)
l=1
bk = r
q
e
| p (x)
2 k! k ,...,k ≥0
r
1
k1 +k2 +...+kr =1
1
= r
2 k!
X
q
Pr
l=1 bl kl
p(k)
r
X
aj kj
j=1
k1 ,...,kr ≥0
k1 +k2 +...+kr =1
!
.
As a result, we state the following theorem.
Theorem 6. For p(x) ∈ P (q | a1 , . . . , ar ; b1 , . . . , br ), when we consider
p(x) =
n
X
bk Ek,q (x | a1 , . . . , ar ; b1 , . . . , br ) ,
k=0
we obtain
1
bk = r
2 k!
X
q
Pr
l=1 bl kl (k)
p
r
X
j=1
k1 ,...,kr ≥0
k1 +k2 +...+kr =1
8
aj kj
!
.
(30)
(31)
3
Acknowledgements
The authors would like to thank the anonymous referee for several helpful comments.
References
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p-adic q-integrals, Computers & Mathematics with Applications 54 (2007) 484–489.
[2] T. Kim, p-adic q-integrals associated with the Changhee–Barnes’ q-Bernoulli polynomials, Integral Transforms Spec. Funct. 15 (2004) 415–420.
[3] T. Kim, Symmetry of power sum polynomials and multivariate fermionic p-adic invariant
integral on Zp , Russ. J. Math. Phys. 16 (2009), 93–96.
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polynomials, Russ. J. Math. Phys. 10 (2003) 91–98.
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umbral calculus, Advances in Difference Equations 2012, 2012:196.
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Ars Combinatoria, to appear.
[7] M. Maldonado, J. Prada, and M. J. Senosiain, Appell bases on sequence spaces, J.
Nonlinear Math. Physics 18, Suppl. 1 (2011) 189–194.
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Adv. Stud. Contemp. Math. 22 (2012), 433–438.
[9] S. Araci, D. Erdal, and J. J. Seo, A study on the fermionic p-adic q-integral representation on Zp associated with weighted q-Bernstein and q-Genocchi polynomials, Abstract
and Applied Analysis, Volume 2011, Article ID 649248.
[10] S. Araci, M. Acikgoz, and J. J. Seo, Explicit formulas involving q-Euler numbers and
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2010 Mathematics Subject Classification: Primary 11S80; Secondary 11B68.
Keywords:
Appell sequence, Sheffer sequence, multivariate q-Euler polynomial, formal
power series.
9
Received January 23 2013; revised versions received June 5 2013; December 7 2013. Published in Journal of Integer Sequences, December 8 2013.
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10