Genocchi polynomials associated with the Umbral algebra

Genocchi polynomials associated with the Umbral algebra

Rahime Dere

^⇑

, Yilmaz Simsek

Department of Mathematics, Faculty of Science, University of Akdeniz, TR-07058 Antalya, Turkey

a r t i c l e i n f o

Dedicated to Professor H. M. Srivastava on the Occasion of his Seventieth Birth Anniversary

Keywords:

Genocchi polynomials

Sheffer sequences and Appell sequences Euler polynomials of higher order Stirling numbers

a b s t r a c t

The aim of this paper is to study on the Genocchi polynomials of higher order onP, the algebra of polynomials in the single variablexover the fieldCof characteristic zero and P, the vector spaces of all linear functional onP. By using the action of a linear functional Lon a polynomialpðxÞSheffer sequences and Appell sequences, we obtain some fundamen- tal properties of the Genocchi polynomials. Furthermore, we give relations between, the first and second kind Stirling numbers, Euler polynomials of higher order and Genocchi polynomials of higher order.

Ó2011 Elsevier Inc. All rights reserved.

1. Introduction

Throughout of this paper, we can use the following notations and definitions, which are given by Roman[3, pp. 1–125].

LetPbe the algebra of polynomials in the single variablexover the field of complex numbers. LetPbe the vector space of all linear functionals onP. LethLjpðxÞibe the action of a linear functionalLon a polynomialpðxÞ. LetFdenote the algebra of formal power series

f tð Þ ¼X¹

k¼0

ak

k!t^k: ð1:1Þ

Such algebra is called Umbral algebra. Eachf2 F defines a linear functional onPand ak¼f tð Þjx^k

ð1:2Þ for allkP0.

The ordero f tðð ÞÞof a power seriesf tð Þis the smallest integerkfor which the coefficient oft^kdoes not vanish. A seriesf tð Þ for whicho f tðð ÞÞ ¼1 will be called a delta series. When we are considering a delta seriesf tð ÞinFas a linear functional we will refer to it as a delta functional.

It is well-known thatt^kjxⁿ

¼n!dn;kwhereddenotes Kronecker symbol. For allf tð ÞinF f tð Þ ¼X¹

k¼0

f tð Þjx^k

k! t^k:

LetfðtÞ,gðtÞbe inF, then we have fðtÞgðtÞjp xð Þ

h i ¼hfðtÞjgðtÞp xð Þi: ð1:3Þ

0096-3003/$ - see front matterÓ2011 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2011.01.078

⇑ Corresponding author.

E-mail addresses:[email protected](R. Dere),[email protected](Y. Simsek).

Contents lists available atScienceDirect

Applied Mathematics and Computation

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / a m c

Fory2Cthe evaluation functional is defined to be the power seriese^yt. By (1.2), we have e^ytjp xð Þ

¼p yð Þ; ð1:4Þ

for allp xð ÞinP. The forward difference functional is the delta functionale^yt1 and e^yt1jp xð Þ

¼p yð Þ pð Þ:0 ð1:5Þ

The Abel functional is the delta functionalte^yt. We have te^ytjp xð Þ

¼p⁰ð Þ:y

The Sheffer polynomials are defined by means of the following generating function X¹

k¼0

skð Þx k! t^k¼ 1

gðtÞe^xt: cf.[3]see also[1,2]).

Roman [3] proved the following theorem which is represented by the Sheffer polynomials (or Sheffer sequences) explicitly:

Theorem 1. Let f tð Þbe a delta series and let g tð Þbe an invertible series. Then there exist a unique sequence snð Þx of polynomials satisfying the orthogonality conditions

gðtÞfðtÞ^kjsnðxÞ D E

¼n!dn;k ð1:6Þ

for all n;kP0.

The sequencesnðxÞin (1.6) is the Sheffer polynomials for pairðgðtÞ;fðtÞÞ, wheregðtÞmust be invertible andfðtÞmust be delta series. The Sheffer polynomials for pairðgðtÞ;tÞis the Appell polynomials or the Appell sequences forgðtÞ.

The Appell polynomials, the Bernoulli polynomials, the Euler polynomials and the Genocchi polynomials belong to the family of the Sheffer polynomials cf.[1–4].

The Sheffer polynomials satisfy the following relations:

snð Þ ¼x gðtÞ¹xⁿ; ð1:7Þ

derivative formula

tsnð Þ ¼x s⁰_nð Þ ¼x nsn1ð Þ;x ð1:8Þ

recurrence formula snþ1ð Þ ¼x xg⁰ð Þt

g tð Þ

snð Þx; ð1:9Þ

expansion theorem h tð Þ ¼X¹

k¼0

h tð Þjskð Þx

h i

k! g tð Þt^k; ð1:10Þ

multiplication theorem, for

a

–0 snð

a

xÞ ¼

a

^{ng tð Þ}

g _a^t snð Þx; ð1:11Þ

and

h tð Þjp axð Þ

h i ¼hh atð Þjp xð Þi: ð1:12Þ

2. Genocchi Polynomials of higher order onF

In this section, by using properties of the Sheffer sequences and also the Appell sequences, we prove many fundamental properties of the Genocchi polynomials of higher order G^ðbÞ_n ðxÞ, which are defined by means of the following generating function:

2t e^tþ1 a

e^xt¼X¹

n¼0

G^ðaÞ_n ðxÞtⁿ

n!; ð2:1Þ

wherej jt <

p

.G^ð1Þ_n ðxÞ ¼GnðxÞdenotes the Genocchi polynomials.

By using (1.7) and (2.1), we arrive at the following Lemma:

Lemma 1

G^ðbÞ_n ðxÞ ¼ 2t e^tþ1 b

xⁿ:

Theorem 2

ðe^tþ1Þ^kjGnðxÞ D E

¼2nðk1Þ!X^k1

j¼0

2^kj1 Sðn1;jÞ ðkj1Þ!;

where GnðxÞand Sðu;

v

Þdenote the Genocchi polynomials and the Stirling numbers of the second kind, respectively.

By Lemma1, we obtain e^tþ1

k

jGnðxÞ D E

¼ e^tþ1k

j 2t e^tþ1xⁿ

:

By using (1.3) and (1.8), we get e^tþ1

k

jGnðxÞ D E

¼2nX^k1

j¼0

ðk1Þ!

ðkj1Þ!2^kj1 ðe^t1Þ^j j! jxⁿ¹

* +

: ð2:2Þ

Setting

Sðn1;jÞ ¼1

j! e^t1j

jxⁿ¹ D E

;

whereSðn1;jÞdenotes the Stirling numbers of second kind cf[3, pp. 59],[5], in (2.2), we arrive at the desired result.

By using (1.8), we arrive at the following lemma:

Lemma 2

tG^{ð Þ}_n^að Þ ¼x nG^{ð Þ}_n1^a ð Þx

Remark 1. A second proof of Lemma2is also obtained from (2.1) by using derivative with respect tox. By Lemma2, one can see that

1

tG^{ð Þ}_n^að Þ ¼x 1

nþ1G^{ð Þ}_nþ1^a ð Þ:x Lemma 3

1 e^tþ1

G^{ð Þ}_n^að Þ ¼x 1

2ðnþ1ÞG^ðaþ1Þ_nþ1 ðxÞ Proof.By (2.1) and Lemma1, we obtain

1 e^tþ1

G^{ð Þ}n^að Þ ¼x 1 e^tþ1

2t e^tþ1 a

xⁿ:

After some calculations in the above equation, we get 1

e^tþ1

G^{ð Þ}_n^að Þ ¼x 1 2t

2t e^tþ1 aþ1

xⁿ:

Using Lemma2, we obtain the desired result. h An integral representation ofD^eta_2t¹jG^ðbÞ_n ðxÞE

is given by the following theorem.

Theorem 3 e^ta1

2t jG^ðbÞ_n ðxÞ

¼1 2

Z a 0

G^ðbÞ_n ðxÞdx:

Proof. By using Lemma2, we have e^ta1

2t jG^ðbÞ_n ðxÞ

¼ e^ta1 2t j 1

nþ1tG^ðbÞ_nþ1ðxÞ

:

By (1.3), we obtain e^ta1

2t jG^ðbÞ_n ðxÞ

¼ 1

nþ1De^at1jG^ðbÞ_nþ1ðxÞE : The desired result follows now from (1.5). h

A recurrence formula forG^ðaÞ_n ðxÞis given by the next theorem.

Theorem 4 (Recurrence formula).

G^ðaþ1Þ_nþ1 ðxÞ ¼2

aðnaþ1ÞG^ðaÞ_nþ1ðxÞ þ ðaxÞðnþ1ÞG^ðaÞ_n ðxÞ :

Proof. Setting g tð Þ ¼ e^tþ1

2t ^a

ð2:3Þ in (1.9), one can obtain

G^ðaÞ_nþ1ðxÞ ¼ xae^ttðe^tþ1Þ t eð ^tþ1Þ

G^ðaÞ_n ðxÞ

¼xG^ðaÞ_n ðxÞ a e^t e^tþ1 ð Þ1

t

G^ðaÞ_n ðxÞ

¼xG^ðaÞ_n ðxÞ þ a

2ðnþ1ÞG^ðaþ1Þ_nþ1 ðxÞ:

Consequently, in the above equations using Lemma3, Remark 1 ande^tG^{ð Þ}_n^að Þ ¼x 2nG^ð_n1^a1Þð Þ x G^{ð Þ}_n^að Þ, we arrive at the desiredx result. h

We now ready to prove multiplication formula for the Genocchi polynomials as follows:

Theorem 5 (Multiplication formula). For every positive odd integer

a

, Gnð

a

xÞ ¼

a

^n1X

a1 j¼0

ð1ÞⁿGn xþj

a

:

Proof. If we substitute (2.3) into (1.11) and leta¼1, then we obtain Gnð

a

xÞ ¼

a

^{n e}

tþ1 2t 2t

a

e^atþ1GnðxÞ ¼

a

^n1e

tþ1 e^atþ1GnðxÞ:

From the above equation, we get Gnð

a

xÞ ¼

a

^n1X

a1 j¼0

ð1Þⁿe^tjaGnðxÞ: By (1.4), we arrive at the desired result. h

Recall that the Euler polynomialsE^{ð Þ}_n^að Þof higher orderx aare defined by the generating series 2

e^tþ1 a

e^xt¼X¹

n¼0

E^ðaÞ_n ðxÞtⁿ n!:

Lemma 4

E^{ð Þ}_n^að Þ ¼x 2 e^tþ1 a

xⁿ:

Proof of Lemma4was given by Roman[3, pp. 101].

The next theorem expresses the Genocchi polynomials in terms of the Euler polynomials and the Stirling numbers of the first kind.

Theorem 6 G^ðaÞ_n ð Þ ¼x X^a

j¼0

X^j

k¼0

X^k

m¼0

a j

j

k 2^kað1Þ^jks k;ð mÞn^mE^ðaÞ_nkð Þ:x

Proof.From Lemma1we find G^ðaÞ_n ð Þ ¼x 1

e^tþ1þ2t1 e^tþ1 a

xⁿ

¼X^a

j¼0

a j

1 e^tþ1 ð Þ^a

X^j

k¼0

j

k 2^kð1Þ^jkt^kxⁿ: Using

t^kxⁿ¼ ðnÞ_kx^nk;

whereðnÞ_k¼nðn1Þ ðnkþ1Þin the above, we have G^ðaÞ_n ð Þ ¼x X^a

j¼0

a j

1 2^a

X^j

k¼0

j

k 2^kð1Þ^jkð Þnk

2 e^tþ1

ð Þ

a

x^nk: By using Lemma4and

ð Þyk¼X^k

m¼0

s k;ð mÞy^m;

wheres k;ð mÞdenotes the Stirling numbers of the first kind, in the above, we obtain the desired result. h

Theorem 7

ðe^tþ1ÞG^{ð Þ}_n^að Þ ¼x 2nG^ð_n1^a1Þð Þ:x Proof.By using (1.7), we obtain

ðe^tþ1ÞG^{ð Þ}_n^að Þ ¼x ð Þ2t 2t ðe^tþ1Þ a1

xⁿ:

By using Lemma1and Lemma2, we obtain the desired result. h

By substitutinga¼1 into Theorem7, we arrive at the following corollary:

Corollary 1

Gnðxþ1Þ þGnð Þ ¼x 2nxⁿ¹: ð2:4Þ

Remark 2. If we seta¼1 in (2.1), then one can arrive at the proof of the above Corollary1as follows:

2te^xt¼X¹

n¼0

Gnðxþ1Þ þGnð Þx

ð Þtⁿ

n!

From the above we arrive at (2.4) cf. ([1,2]).

Acknowledgements

The present investigation was supported by theScientific Research Project Administration of Akdeniz University.

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