Modification and unification of the Apostol-type numbers and polynomials and their applications

Modification and unification of the Apostol-type numbers and polynomials and their applications

Hacer Ozden

^a,⇑

, Yilmaz Simsek

^b

aUniversity of Uludag, Faculty of Arts and Science, Department of Mathematics, Bursa, Turkey

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

a r t i c l e i n f o

Keywords:

Bernoulli numbers and polynomials Euler numbers and polynomials Genocchi Bernoulli numbers and polynomials

Stirling numbers Catalan numbers

a b s t r a c t

In this paper, we construct generating functions for modification and unification of the Apostol-type polynomialsY^ð_n;b^vÞðx;k;a;bÞof order

v

. By using these generating functions, we derive many new identities related to the generalized Stirling type numbers of the second kind, array-type polynomials, Eulerian polynomials and the modification and unification of the Apostol-type polynomials and numbers. We give many applications related to these numbers and polynomials and PDEs.

Ó2014 Elsevier Inc. All rights reserved.

1. Introduction, definitions and preliminaries

Throughout our paper, we use the following standard notation:N:¼ f1;2;3;. . .gdenotes the set of natural numbers, N0:¼N[ f0g;Zdenotes the set of integers,Qdenotes the set of rational numbers,Rdenotes the set of real numbers,R^þ denotes the set of positive real numbers andCdenotes the set of complex numbers. We also assume that logz denotes the principal branch of the multi-valued function logzwith the imaginary partIðlogzÞconstrained by

p

<IðlogzÞ6

p

. For all 06k6n, letðnÞ_k¼k! n

k (for example, see[31,32]).

Ozden[14]defined the following generating functions which are related to the unification of the Bernoulli, Euler and Genocchi polynomials:

g_bðx;t;k;a;bÞ:¼2^1kt^ke^tx b^be^ta^b¼X¹

n¼0

Yn;bðx;k;a;bÞtⁿ

n!: ð1Þ

Note that forx¼1, Eq.(1)reduces to the generating functions for the unification of the Bernoulli, Euler and Genocchi numbers. In(1)we assume that ifb¼a, thenjtj<2

p

and ifb–a;k2N0;a;b2Cn f0g, thenjtj<blog ^b_a . Also, by using the special values ofa;b;kandbin(1), the polynomialsYn;bðx;k;a;bÞprovides us with a generalization and unification of the Apostol–Bernoulli polynomials, Apostol–Euler polynomials and Apostol–Genocchi polynomials, respectively:

Bnðx;bÞ ¼Yn;bðx;1;1;1Þ;

Enðx;bÞ ¼Yn;bðx;0;1;1Þ

http://dx.doi.org/10.1016/j.amc.2014.03.004 0096-3003/Ó2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author.

E-mail addresses:[email protected](H. Ozden),[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

and

Gnðx;bÞ ¼Yn;bðx;1;1;1Þ

(cf.[2–34]. Furthermore, for the classical Bernoulli polynomialsBnðxÞ, the classical Euler polynomialsEnðxÞand the classical Genocchi polynomialsGnðxÞ, we have

BnðxÞ ¼Bnðx;1Þ; EnðxÞ ¼Enðx;1Þ and

GnðxÞ ¼Gnðx;1Þ

(cf.[1–34]and the references cited in each of these earlier works). Forx¼0, we have the classical Bernoulli numbersBn, the classical Euler numbersEnand the classical Genocchi numbersGn, we have

Bn¼Bnð0Þ;

En¼Enð0Þ and

Gn¼Gnð0Þ

(cf.[1–36]and the references cited in each of these earlier works).

In analogy with the generating functions introduced by Ozden[14]and also[34]for the unification of the Apostol-type numbers and polynomials, we define generating functions for these polynomials. Now, we modify and unify(1)as follows:

Leta;b2R^þ(a–b). Thenmodification and unification of the Apostol-type polynomialsY^ð_n;b^vÞðx;k;a;bÞof order v are defined by means of the following generating function:

Mk;vðt;x;a;b;bÞ ¼ t^k2^1k bb^ta^t

!v b^xt¼X¹

n¼0

Y^ð_n;b^vÞðx;k;a;bÞtⁿ

n!; ð2Þ

where

tln b a þlnb

<2

p

andx2R. We observe that

Y^ð_n;b^vÞð0;k;a;bÞ ¼Y^ð_n;b^vÞðk;a;bÞ;

which denotesmodification and unification of the Apostol-type numbers of orderv. Therefore, the numbersY^ð_n;b^vÞðk;a;bÞare defined by means of the following generating functions:

t^k2^1k bb^ta^t

!v

¼X¹

n¼0

Y^ð_n;b^vÞðk;a;bÞtⁿ

n!: ð3Þ

Remark 1.1. By substitutingk¼1 into(2), we have known results of[33]

B^ðn^vÞðx;b;k;a;bÞ ¼Y^ð_n;b^vÞðx;1;a;bÞ (cf. see also[27,34]).

2. Some properties of the numbersY^ð_n;b^vÞðk;a;bÞand the polynomialsY^ð_n;b^vÞðx;k;a;bÞ

Here, by using generating functions, we give some properties of the numbers Y^ð_n;b^vÞðk;a;bÞ and the polynomials Y^ð_n;b^vÞðx;k;a;bÞ.

By using(1) and (2), we get

Mk;1ðt;x;a;b;bÞ ¼ ln b a k

g_b xlnblna ln ^b_a ;tln b

a ;k;1;1

! :

Thus, we get

Yn;bðx;k;a;bÞ ¼ ln b a nk

Yn;b

xlnblna ln ^b_a ;k;1;1

!

;

whereYn;bðx;k;a;bÞdenotes the Apostol-type polynomials (cf.[14,16,32]; see also the references cited in each of these earlier works).

M1;vðt;x;1;e;1Þ ¼X¹

n¼0

B^ðn^vÞðx;bÞtⁿ n!; where

B^ðn^vÞðx;bÞ ¼Yn;bðx;1;1;eÞ

denotes the Apostol–Bernoulli polynomials of order

v

^(cf.[2–34]and the references cited in each of these earlier works).

M0;vðt;x;1;e;1Þ ¼X¹

n¼0

ð1Þ^vE^ð_n^vÞðxÞtⁿ

n!; ð4Þ

where

ð1Þ^vE^ð_n^vÞðxÞ ¼Yn;1ðx;k;1;eÞ

denotes the Euler polynomials of order

v

^(cf.[2–34]; see also the references cited in each of these earlier works).

Substitutingx¼0 and

v

¼1 into(2)and using the Umbral Calculus convention, we derive a recurrence relation for the modification and unification of the Apostol-type numbers,Yn;bðk;a;bÞ. Therefore, we set

t^k2^1k¼be^tlnbe^tlna e^Ybðk;a;bÞt: From the above equation, we obtain

t^k2^1k¼ bX¹

n¼0

tlnb ð Þⁿ

n! X¹

n¼0

tlna ð Þⁿ

n!

!X¹

n¼0

Yn;bðk;a;bÞtⁿ n!: Therefore

t^k2^1k¼X¹

n¼0

bXⁿ

j¼0

n

j ðlnbÞ^njYj;bðk;a;bÞ Xⁿ

j¼0

n

j ðlnaÞ^njYj;bðk;a;bÞ

!tⁿ n!:

By comparing the coefficients oftⁿon both sides of the above equation, we arrive at the following theorem:

Theorem 2.1.The following recurrence relation holds true:

bðYbðk;a;bÞ þlnbÞⁿYbðk;a;bÞ þlnan

¼ 2^1kk! n¼k;

0 n–k:

(

where Y bðk;a;bÞm

is replaced conventionally by Ym;bðk;a;bÞ.

By substitutingn¼0 intoTheorem 2.1, we find that Y0;bð0;a;bÞ ¼ 2

b1 and

Y0;bð1;a;bÞ ¼0;

whereb–1. By substitutingn¼1 intoTheorem 2.1, we can easily calculate the numbersY1;bðk;a;bÞas follows:

Ifk¼1, we have Y1;bð1;a;bÞ ¼ 1

b1: Ifk¼0, we have

Y1;bð0;a;bÞ ¼2ðb1Þ 2ðblnblnaÞ ðb1Þ² :

Consequently, applyingTheorem 2.1, we can easily calculate all the numbersYn;bðk;a;bÞ.

By using(3), we get X¹

n¼0

Y^ð_n;b^vþpÞðk;a;bÞtⁿ n!¼X¹

n¼0

Y^ð_n;b^vÞðk;a;bÞtⁿ n!

X¹

n¼0

Y^ðpÞ_n;bðk;a;bÞtⁿ n!: Therefore

X¹

n¼0

Y^ð_n;b^vþpÞðk;a;bÞtⁿ n!¼X¹

n¼0

Xⁿ

j¼0

n

j Y^ð_j;b^vÞðk;a;bÞY^ðpÞ_nj;bðk;a;bÞ

!tⁿ n!:

By comparing the coefficients of^t_n!ⁿon both sides of the above equation, we arrive at the following theorem:

Theorem 2.2. The following recurrence relation holds true:

Y^ð_n;b^vþpÞðk;a;bÞ ¼Xⁿ

j¼0

n

j Y^ð_j;b^vÞðk;a;bÞY^ðpÞ_nj;bðk;a;bÞ:

By applying Theorem 2.2, we can calculate all the numbers Y^ð_n;b^vþpÞðk;a;bÞ. For example, if we substitute n¼1 and

v

¼p¼1 intoTheorem 2.2, we get Y^ð2Þ_1;bðk;a;bÞ ¼X¹

j¼0

1

j Yj;bðk;a;bÞY1j;bðk;a;bÞ: Ifk¼1, we have

Y^ð2Þ_1;bð1;a;bÞ ¼0:

Ifk¼0, we have

Y^ð2Þ_1;bð0;a;bÞ ¼8ðb1Þ 8ðblnblnaÞ ðb1Þ³ :

Thus we are ready to generalizeTheorem 2.2as follows:

Theorem 2.3. Let n;j₁;j₂;. . .;j_r2N0. Then we have Y^ð_n;b^{v1þv2þþvrÞ}ðk;a;bÞ ¼^j1þj2X^þþjr¼n

j₁;j₂;...;j_r¼0

j₁þj₂þ þj_r

ð Þ!

j₁!j₂! j_r! Y^r

c¼1

Y^ð_j^vcÞ

c;bðk;a;bÞ:

Proof. By using(3), we get

t^k2^1k bb^ta^t

!v1þv2þþvr

¼X¹

n¼0

Y^ð_n;b^{v1þv2þþvrÞ}ðk;a;bÞtⁿ n!:

By usingTheorem 2.2and mathematical induction, we complete proof of theorem. h By using(2) and (3), we easily arrive at the following theorem:

Theorem 2.4. Let n2N0. The we have Y^ð_n;b^vÞðx;k;a;bÞ ¼Xⁿ

j¼0

n

j x^njðxlnbÞ^njY^ð_j;b^vÞðk;a;bÞ:

3. PDEs for the generating functions

In this section, we give PDEs for the generating functions. By using these equations, we derive some derivative formulas and recurrence relations of themodification and unification of the Apostol-type polynomials of order

v

. Taking derivative of(2), with respect tox, we obtain the following PDE for the generating function:

@^m

@x^mMk;vðt;x;k;a;b;bÞ ¼ ðtlnbÞ^mMk;vðt;x;k;a;b;bÞ: ð5Þ Taking derivative of(2), with respect tob, we obtain the following PDE for the generating function:

@

@bMk;vðt;x;k;a;b;bÞ ¼

v

2^1kt^kMk;vþ1ðt;xþ1;k;a;b;bÞ ð6Þ

or

@

@bMk;vðt;x;k;a;b;bÞ ¼

v

2^1kt^kMk;v¹ðt;xþ1;k;a;b;bÞMk;2ðt;k;a;b;bÞ: ð7Þ Taking derivative of(2), with respect tot, we obtain the following PDE for the generating function:

@

@tMk;vðt;x;k;a;b;bÞ ¼k

v

t Mk;vðt;x;k;a;b;bÞ blnb

2^1kt^kMk;vþ1ðt;xþ1;k;a;b;bÞ þ lna

2^1kt^ka^tMk;v^þ1ðt;x;k;a;b;bÞ þxlnbMk;vðt;x;k;a;b;bÞ: ð8Þ By using the above PDEs, we derive the following theorems.

Theorem 3.1.Let m and n be positive integers with nPm. Then we have

@^m

@x^mY^ð_n;b^vÞðx;k;a;bÞ ¼m! n m

lnb

ð Þ^mY^ð_nm;b^vÞ ðx;k;a;bÞ

Proof. By using(5)with(2), we obtain X¹

n¼0

@^m

@x^mY^ð_n;b^vÞðx;k;a;bÞtⁿ n!¼X¹

n¼0

m! n m

ðlnbÞ^mY^ð_nm;b^vÞ ðx;k;a;bÞtⁿ n!:

By comparing the coefficients of^t_n!ⁿon both sides of the above equation, we obtain the result. h

Theorem 3.2. Let v and n be positive integers. Then we have

@

@bY^ð_n;b^vÞðx;k;a;bÞ ¼ 2^1k

v

nþk k

Y^ð_nþk;b^vþ1Þðxþ1;k;a;bÞ

or

@

@bY^ð_n;b^vÞðx;k;a;bÞ ¼ 2^1k

v

nþk k X^nþk

j¼0

nþk j

Y^ð_j;b^v1Þðxþ1;k;a;bÞY^ð2Þ_nþkj;bðk;a;bÞ: ð9Þ

Proof. By using(6)with(2), we obtain X¹

n¼0

@

@bY^ð_n;b^vÞðx;k;a;bÞtⁿ n!¼

v

2^1k X¹

n¼0

Y^ð_n;b^vþ1Þðxþ1;k;a;bÞt^nk n! :

Therefore X¹

n¼0

@

@bY^ð_n;b^vÞðx;k;a;bÞtⁿ

n!¼ 2^1k

v

^X

1

n¼0

nþk k ¹

Y^ð_nþk;b^vþ1Þðxþ1;k;a;bÞtⁿ n!:

By comparing the coefficients of^t_n!ⁿon both sides of the above equation, we obtain the result. h By using(7), we easily obtain the assertion(9)ofTheorem 3.2.

Theorem 3.3(Convolution recurrence relation). Let n2N0. The following relationship hold true:

nþ1

v

^k

nþ1 Y^ð_nþ1;b^vÞ ðx;k;a;bÞ þ2^k1blnb nþk

k

Y^ð_nþk;b^vþ1Þðxþ1;k;a;bÞ

¼xlnbY^ð_n;b^vÞðx;k;a;bÞ þ 2^k1 nþk

k X^nþk

j¼0

nþk j

ðlnaÞ^nþkþ1jY^ð_j;b^vþ1Þðxþ1;k;a;bÞ: ð10Þ

Proof. By using(8)with(2), after some elementary calculations, we obtain X¹

n¼1

Y^ðn;b^vÞðx;k;a;bÞ tⁿ¹

ðn1Þ!xlnbX¹

n¼0

Y^ðn;b^vÞðx;k;a;bÞtⁿ n!¼

v

kX¹

n¼0

Y^ðn;b^vÞðx;k;a;bÞtⁿ¹

n! 2^k1blnbX¹

n¼0

Y^ðn;b^vþ1Þðx;k;a;bÞt^nk n!

þX¹

n¼0

2^k1 nþk

k X^nþk

j¼0

nþk j

ðlnaÞ^nþkþ1jY^ð_j;b^vþ1Þðxþ1;k;a;bÞtⁿ n!:

By comparing the coefficients of^t_n!ⁿon both sides of the above equation, we obtain the result. h Substitutinga¼k¼

v

^{¼1 andb¼einto}(10), we obtain the following corollary:

Corollary 3.1. Let n2N0. The following relationship hold true:

bB^ð2Þnþ1ðxþ1;bÞ ¼ ðnþ1ÞxBnðx;bÞ nBnþ1ðx;bÞ: ð11Þ Substitutingb¼1 into(11)and usingTheorem 2.2, we obtain the following corollary:

Corollary 3.2. Let n2N0. The following relationship hold true:

X^nþ1

j¼0

nþ1 j

ðjx^j1þBjðxÞÞBnþ1j¼xðnþ1ÞBnðxÞ nBnþ1ðxÞ: ð12Þ Substitutingx¼0 into(12), one can easily arrive at the following convolution recurrence relation for the Bernoulli numbers:

Corollary 3.3. Let n2N0. The following relationship hold true:

Bnþ1þ1 2Bnþ 1

nþ1 X^nþ1

j¼2

nþ1 j

BjBnþ1j¼0; ð13Þ

where B0¼1.

Remark 3.1. A convolution recurrence relation in(13)give us modification of the Euler’s convolution recurrence relation for the Bernoulli numbers:

1 n

Xⁿ

j¼1

n

j BjBnj¼ BnBn1; wherenP1 (cf.[7,8,6,27,30]).

Substitutinga¼

v

¼1;k¼0;b¼ 1 andb¼einto(10), we obtain the following corollary:

Corollary 3.4.

E^ð2Þ_n ðxþ1Þ ¼2Enþ1ðxÞ 2xEnðxÞ: ð14Þ

By(14), we have Xⁿ

j¼0

n

j Ejðxþ1ÞEnj¼2Enþ1ðxÞ 2xEnðxÞ:

Substituting the following well-known identity Enðxþ1Þ ¼2xⁿEnðxÞ

into the above equation, we easily obtain the following convolution recurrence relation for the Euler polynomials:

Xⁿ

j¼0

n

j ð2xⁿEnðxÞÞEnj¼2Enþ1ðxÞ 2xEnðxÞ:

Substitutingx¼0 into the above equation, we arrive at the known result due to convolution recurrence relation for the Euler numbers:

Xⁿ

j¼1

n

j EjEnj¼En2Enþ1

(cf.[7,8,6,27,30]).

4. Identities related to the polynomialsY^ð_n;b^vÞðx;k;a;bÞandb-Stirling type numbers

By using an approach similar to that of Srivastava[27]as well as some well-known properties of the Stirling numbers of the second kind, we can derive some known and some new identities and formulas for themodification and unification of the Apostol-type polynomials Y^ð_n;b^vÞðx;k;a;bÞof order v and theb-Stirling type numbers.

We need the following generating function for generalizedb-Stirling type numbers of the second kind, related to nonneg- ative positive real parameters.

Definition 4.1 (cf.[25, p. 3, Definition 2.1]). Leta;b2R^þ(a–b),b2Cand

v

2N0. The generalizedb-Stirling type numbers of the second kindSðn;

v

^;a;b;bÞare defined by means of the following generating function:

fS;vðt;a;b;bÞ ¼bb^ta^tv

v

^{! ¼}

X¹

n¼0

Sðn;

v

^;a;b;bÞ_n!^{tn: ð15Þ}

By settinga¼1 andb¼ein(15), we have theb-Stirling numbers of the second kind Sðn;

v

^{;1;e;kÞ ¼Sðn;}

v

^;kÞ;

which are defined by means of the following generating function:

be^t1 ð Þ^v

v

^{! ¼}

X¹

n¼0

Sðn;

v

^;bÞtⁿ n!;

(cf.[9,27,30]). Substitutingb¼1 into above equation, we have the Stirling numbers of the second kindSðn;

v

^{;1Þ ¼Sðn;}

v

^Þ(cf.

[7,9,19,27,29,30]).

In[25, p. 3, Theorem 2.2], Simsek gave the following formulas for the generalizedb-Stirling type numbers of the second kindSðn;

v

^;a;b;bÞ:

Theorem 4.1.We have Sðn;

v

^{;a;b;bÞ ¼}

v

^1!

X^v

j¼0

ð1Þ^j

v

j b^vjðjlnaþ ð

v

^jÞlnbÞn ð16Þ

and

Sðn;

v

^;a;b;bÞ ¼ 1

v

^!

X^v

j¼0

ð1Þ^vj

v

j b^jðjlnbþ ð

v

jÞlnaÞⁿ: ð17Þ

Note that by settinga¼1 andb¼ein the assertions(16)ofTheorem 4.1, we have the following result:

Sðn;

v

^{;bÞ ¼}

v

^1!

X^v

j¼0

v

j b^vjð1Þ^jð

v

^jÞn:

The above relation has been studied by Srivastava[27]and Luo and Srivastava[9]. By settingb¼1 in the above equation, we have the following well-known result for the classical Stirling numbers of the second kind:

Sðn;

v

^{Þ ¼}

v

^1!

X^v

j¼0

v

j ð1Þ^jð

v

^jÞn

(cf.[1,6,5,7,9,19,22,25,27,30]).

In[25], Simsek constructed generating function of thegeneralized array type polynomialsas follows: Leta;b2R^þ, (a–b), k2Cand

v

2N0.

g_vðx;t;a;b;bÞ ¼ 1

v

^{! bb}

ta^t v

b^xt¼X¹

n¼0

Sⁿ_vðx;a;b;bÞtⁿ

n!: ð18Þ

The following definition provides a natural generalization and unification of the array polynomials:

Definition 4.2 [25]. Leta;b2R^þ(a–b),x2R;k2Cand

v

^2N0. The generalized array type polynomialsSⁿvðx;a;b;kÞcan be defined by

Sⁿ_vðx;a;b;kÞ ¼1

v

^!

X^v

j¼0

ð1Þ^vj

v

j k^jlna^vjb^xþjn

: ð19Þ

Remark 4.1. The polynomialsSⁿ_vðx;a;b;kÞmay be also called generalizedk-array type polynomials. By substitutingx¼0 into(19), we arrive at(17):

Sⁿ_vð0;a;b;kÞ ¼Sðn;

v

^;a;b;kÞ:

Settinga¼k¼1 andb¼ein (19), we have Sⁿ_vðxÞ ¼ 1

v

^!

X^v

j¼0

ð1Þ^vj

v

j ðxþjÞⁿ;

a result due to Chang and Ha[3, Eq-(3.1)], Simsek[22]. It is easy to see that S⁰₀ðxÞ ¼Sⁿ_nðxÞ ¼1;

Sⁿ₀ðxÞ ¼xⁿ and for

v

^>n,

Sⁿ_vðxÞ ¼0;

cf.[3, Eq-(3.1)].

By using(2) and (15), we derive the following functional equation:

2^vðk1Þ

v

^!

t^kv Mk;vðt;x;a;b;bÞfS;vðt;a;b;bÞ ¼b^tx: ð20Þ

Theorem 4.2. Let n;k;

v

2N. Thus we have

X

nþkv

l¼0

nþk

v

l

Sðnþk

v

l;

v

^;a;b;bÞY^ð_l;b^vÞðx;k;a;bÞ ¼ nþk

v

k

v

xlnb ð Þⁿ

2^vðk1Þ

v

^{!: ð21Þ}

Proof. By using(20), we get

2^vð1kÞ

v

^!

X¹

n¼0

xlnb ð Þⁿtⁿ

n!¼t^kvX¹

n¼0

Sðn;

v

^;a;b;bÞtⁿ n!

X¹

n¼0

Y^ð_n;b^vÞðx;k;a;bÞtⁿ n!: Therefore

2^vð1kÞ

v

^!

X¹

n¼0

xlnb ð Þⁿtⁿ

n!¼X¹

n¼0

X

nþkv

l¼v

nþk

v

l

nþk

v

k

v

S^ðnþk

v

^l;

v

^;a;b;bÞYðl;b^vÞðx;k;a;bÞtⁿ n!:

By comparing the coefficients of^t_n!ⁿon both sides of the above equation, we obtain the result. h

Remark 4.2. Substitutingk¼a¼1;b¼einto(21), we obtain known result due to[27, p. 420, Theorem 15]see also[9].

Substitutingx¼0 into(21), we have the following results:

Corollary 4.1. If n¼0, then we have

X^kv

l¼0

k

v

l

Sðk

v

l;

v

^;a;b;bÞY^ð_l;b^vÞðk;a;bÞ ¼1 and if n–0, then we obtain

X

nþkv

l¼0

nþk

v

l

Sðnþk

v

l;

v

^;a;b;bÞY^ð_l;b^vÞðk;a;bÞ ¼0: ð22Þ

Remark 4.3. Fork¼a¼1;b¼e, Eq.(22)reduces to the known result given recently by Srivastava[27, p. 417, Theorem 13].

Corollary 4.2. We have Y^ð_nþk^vÞ_v;bðk;a;bÞ ¼

v

^!

ðb1Þ^v X

nþkv1

l¼0

nþk

v

l

S^ðnþk

v

^l;

v

^;a;b;bÞYðl;b^vÞðk;a;bÞ:

Proof. By(21), we have

0¼^nþkX^v1

l¼0

nþk

v

l

Sðnþk

v

l;

v

^;a;b;bÞY^ð_l;b^vÞðk;a;bÞ þY^ð_nþk^vÞ_v;bðk;a;bÞSð0;

v

^;a;b;bÞ:

By using the formula(16), we have Sð0;

v

^{;a;b;bÞ ¼ðb}

v

^!1Þv:

Substituting these numbers into the above equation, we obtain the desired result. h

Remark 4.4. By substituting

v

¼a¼k¼b¼1;b¼eintoCorollary 4.2, we obtain B^ðnþ^vÞvðbÞ ¼

v

^!

ðb1Þ^v X

nþv¹

l¼0

nþ

v

l

S nð þ

v

l;

v

^;bÞB^ð_l^vÞðbÞ

(cf.[27, p. 418, Eq-(7.7)]). By same method of[27], for

v

^{¼a¼k¼b¼1;b¼eandSðn;1Þ ¼1, Eq.(22)}reduces to the follow- ing well known results for the classical Bernoulli numbers:B0¼1 and

Bn¼ 1 nþ1

Xⁿ¹

l¼0

nþ1 l

Bl

(cf. see for detail[27, p. 418, Eq-(7.8)]).

We now define modification and unification of the Apostol-type polynomials of order

v

^;Yðn;b^vÞðx;k;a;bÞby means of the following generating function

Mk;vðt;x;a;b;bÞ ¼ bb^ta^t t^k2^1k

!v b^xt¼X¹

n¼0

Y^ð_n;b^vÞðx;k;a;bÞtⁿ

n!: ð23Þ

In work of Srivastava, we know that the Apostol-type polynomials of order

v

are related to the Stirling-type numbers[27, p.

417].

Theorem 4.3.We have Y^ð_n;b^vÞðx;k;a;bÞ ¼

v

^!2vðk1Þ

ðk

v

Þ!

nþk

v

k

v

^1nþkX^v

j¼0

nþk

v

j

S^ðj;

v

^{;a;b;bÞxnþkvj:}

First Proof of Theorem 4.3. By combing(23)with(15), we get X¹

n¼0

Y^ð_n;b^vÞðx;k;a;bÞtⁿ

n!¼

v

^{!2ðk1ÞvtkvX1}

n¼0

Sðn;

v

^;a;b;bÞtⁿ n!

X¹

n¼0

xlnb ð Þⁿtⁿ

n!: Therefore

X¹

n¼0

Y^ð_n;b^vÞðx;k;a;bÞtⁿ n!¼X¹

n¼0

v

^!2vðk1Þ

k

v

ð Þ! nþk

v

k

v

0 BB B@

1 CC CA

X

nþkv

j¼0

nþk

v

j

S^ðj;

v

^{;a;b;bÞxnþkvjt}

n

n!:

Thus by comparing the coefficients oftⁿon both sides of the above equation, we arrive at the desired result. h

Theorem 4.4. We have

Y^ð_n;b^vÞðx;k;a;bÞ ¼

v

^!2vðk1Þ

k

v

ð Þ! nþk

v

k

v

S^nþk_v ^vðx;a;b;kÞ:

ð24Þ

Proof. By combining(23)with(18), we get X¹

n¼0

Y^ð_n;b^vÞðx;k;a;bÞtⁿ n!¼X¹

n¼0

v

^!2vðk1Þ

k

v

ð Þ! nþk

v

k

v

S^nþk_v ^vðx;a;b;kÞtⁿ n!:

Thus by comparing the coefficients oftⁿon the both sides of the above equation, we arrive at the desired result. h

First Proof of Theorem 4.3. By combing(23)with(15), we get

Y^ð_n;b^vÞðx;k;a;bÞ ¼

v

^!2vðk1Þ

k

v

ð Þ! nþk

v

k

v

^nþkX^v

j¼0

nþk

v

j

Sðj;

v

^;a;b;kÞðxlnbÞ^nþkvj:

Thus the proof is completed. h

Remark 4.5. If we substitutea¼k¼1;b¼eintoTheorem 4.3, we have known result in[27, p. 419, Theorem 14].

By using(24) and (19), we can compute some values of the polynomialsY^ð_n;b^vÞðx;k;a;bÞas follows (see[25]):

Y^ð0Þ_n;bðx;k;a;bÞ ¼Sⁿ0ðx;a;b;bÞ;

Y^ð_0;b^vÞðx;k;a;bÞ ¼2^vðk1ÞS⁰_vðx;a;b;bÞ and

Y^ð1Þ_1;b ðx;k;a;bÞ ¼ 2^ðk1Þ k! 1þk

k

S¹1ðx;a;b;kÞ ¼ 2^ðk1Þ k! 1þk

k

ðlnaxlnbþbðxþ1ÞlnbÞ:

If we substitutex¼0 intoTheorem 4.3, we get the following result.

Corollary 4.3. We have Y^ð_n;b^vÞðk;a;bÞ ¼

v

^!2vðk1Þ

ðk

v

^Þ!

nþk

v

k

v

1

Sðnþk

v

^;

v

^;a;b;bÞ: ð25Þ

Remark 4.6. By substituting

v

^{¼k¼n, into(25)we get}

Y^ðnÞ_n;b ðn;a;bÞ ¼n!2^nðn1Þ ðn²Þ!

nþn² n²

!1

Snþn²;n;a;b;b :

By substituting

v

^{¼n;a¼k¼b¼1 andb¼einto}(25), we have

B^ðnÞ_n ¼ 2n n ¹

Sð2n;nÞ; ð26Þ

Sð2n;nÞdenotes the Stirling numbers of the second kind[27, p. 419, Eq-(7.18)].

Remark 4.7. In(26), we easily see that Sð2n;nÞ ¼ ðnþ1ÞCnB^ðnÞ_n ;

whereCndenotes the Catalan numbers (cf.[4]) which are defined by Cn¼ 1

nþ1 2n

n

:

In[34], Srivastava et al. defined the generalized Apostol-type Genocchi polynomials as follows:

2t bb^tþa^t v

c^xt¼X¹

n¼0

G^ðn^vÞðx;b;a;b;cÞtⁿ n!;

(cf. see also[33],[30]). In the next theorem, we give relationship between the Stirling-type numbers and the generalized Apostol-type Genocchi polynomials of order

v

^.

Theorem 4.5.We have

Snþ

v

^;k

v

^;a2;b2;b2¼2^v nþ

v v

Xⁿ

j¼0

n

j Sðjþ

v

^;

v

^;a;b;bÞG^ð_nj^vÞðb;a;bÞ:

Proof. By using(15), we get the following functional equation:

1 2t

ð Þ^vfS;vð2t;a²;b²;b²Þ ¼fS;vðt;a;b;bÞfG;vðt;a;b;bÞ; ð27Þ where

fG;vðt;a;b;bÞ ¼ 2t bb^tþa^t v

¼X¹

n¼0

G^ðn^vÞðb;a;bÞtⁿ n!;

G^ðn^vÞðb;a;bÞdenotesb-Genocchi numbers of order

v

, related to nonnegative real parameters.

By using(27), we obtain 1

2t ð Þ^v

X¹

n¼0

Sn;

v

^;a2;b2;bt_n!ⁿ¼X¹

n¼0

Sðn;

v

^;a;b;bÞtⁿ n!

X¹

n¼0

G^ðn^vÞðb;a;bÞtⁿ n!:

Therefore X¹

n¼0

2^v nþ

v v

¹

Snþ

v

^;k

v

^;a2;b2;b2t_n!ⁿ¼X¹

n¼0

Xⁿ

j¼0

n

j Sðjþ

v

^;

v

^;a;b;bÞG^ð_nj^vÞðb;a;bÞtⁿ n!:

Thus by comparing the coefficients oftⁿon both sides of the above equation, we arrive at the desired result. h

5. Identities related to the Eulerian type numbers and the Stirling type numbers

In this section, by using generating functions, we derive identities related to the Eulerian type numbers and the Stirling type numbers.

In [25,24, p.10], Simsek constructed generating functions of the Eulerian type polynomials of higher order H^ðmÞn ðx;u;a;b;c;kÞas follows:

F^ðmÞ_k ðt;x;u;a;b;cÞ ¼ða^tuÞ^mc^xt kb^tu

m¼X¹

n¼0

H^ðmÞn ðx;u;a;b;c;bÞtⁿ

n!; ð28Þ

wherem2Nand

H^ðmÞn ð0;u;a;b;c;bÞ ¼H^ðmÞn ðu;a;b;bÞ;

denotes the Eulerian type numbers of higher order. By combining(15) with the above generating function, we derive the following functional equation:

1 u1 m

m! b^xt¼fS;m t;1;b;b u X¹

n¼0

H^ðmÞn ðx;u;1;b;b;bÞtⁿ n!: By using this functional equation, we get

1 u1 m

m!

X¹

n¼0

ðxlnbÞⁿtⁿ n!¼X¹

n¼0

S n;m;1;b;b u tⁿ

n!

X¹

n¼0

H^ðmÞn ðx;u;1;b;b;kÞtⁿ n!: By using Cauchy product in the above equation, we obtain

1 u1 m

m!

X¹

n¼0

ðxlnbÞⁿtⁿ n!¼X¹

n¼0

Xⁿ

l¼0

n

l S l;m;1;b;b u

H^ðmÞnlðx;u;1;b;b;bÞ

!tⁿ n!:

By comparing the coefficients of^t_n!ⁿon both sides of the above equation, we arrive at the following theorem:

Theorem 5.1.

1 u1 m

m! ðxlnbÞⁿ¼Xⁿ

l¼0

n

l S l;m;1;b;b u

H^ðmÞ_nlðx;u;1;b;b;bÞ: ð29Þ Substitutingb¼e;b¼1 into(29), we have the following result:

Corollary 5.1.

1 u1 m

m! xⁿ¼Xⁿ

l¼0

n

l S l;ð mÞH^ðmÞ_nlðx;uÞ:

We now derive the following functional equations:

Mk;vðt;k;a;b;bÞ ¼ ð1Þ^vF^ð₁^vÞ t;x;u;a;a b;b

G^ð_Y^vÞðt;a;bÞ; ð30Þ

where

G^ð_Y^vÞðt;a;bÞ ¼ 1 ba^t1 v

¼X¹

n¼0

y^ð_n^vÞðb;aÞtⁿ n!

andaP1 (cf.[25, p-21, Eq-(37)]) Mk;vðt;x

v

^;a;b;bÞ ¼2^kvMk;v 2t;x

2; ffiffiffiffiffiffi pab

;b;b

ð31Þ and

Mk;vðt;

v

^{x;a;b;bÞ ¼ ð1Þvð1kÞbvatvMk;}vt;x;a;b;b¹

: ð32Þ

Theorem 5.2. Let nk

v

^P0. The following identity holds true:

ð1Þ^v2^vðk1Þ nþk

v

k

v

k

v

ð Þ!

Y^ð_n;b^vÞðxþ

v

^{;k;a;bÞ ¼X}

nkv

l¼0

nk

v

l

H^ð_l^vÞ x;b;a;a b;b;1

y^ð_nk^vÞ_vlðb;aÞ:

ð33Þ

Proof. By combining(30)with(2) and (28), we get X¹

n¼0

Y^ð_n;b^vÞðx;k;a;bÞtⁿ

n!¼ ð1Þ^v2^vð1kÞt^kvX¹

n¼0

H^ðn^vÞ x;b;a;a b;b;1 tⁿ

n!

X¹

n¼0

y^ð_n^vÞðb;aÞtⁿ n!: Therefore

ð1Þ^v2^vðk1ÞX¹

n¼0

Y^ð_n;b^vÞðx;k;a;bÞtⁿ n!¼X¹

n¼0

Xⁿ

l¼0

n

l H^ðl^vÞ x;b;a;a b;b;1

y^ð_nl^vÞðb;aÞt^nkv n! :

By comparing the coefficients of^t_n!ⁿon both sides of the above equation, we arrive at the desired result. h

Remark 5.1. Substitutinga¼

v

^{¼k¼1 andb¼einto}(33), we have 1

nþ1Bnðxþ1;bÞ ¼ 1 b1

Xⁿ¹

l¼0

n1 l

Hlðx;bÞ:

By combining(31)with(2), after some elementary calculation, we arrive at the following theorem:

Theorem 5.3.

Y^ð_n;b^vÞðx

v

^{;k;a;bÞ ¼2nkvYð}n;b^vÞ

x 2;k; ffiffiffiffiffiffi

pab

;b

:

By combining(32)with(2), after some elementary calculation, we arrive at the following theorem:

Theorem 5.4.The following identity holds true:

Y^ð_n;b^vÞð

v

^{x;k;a;bÞ ¼ð1Þv}

ð1kÞþn

b^v Xⁿ

j¼0

n

j ^ð

v

^lnaÞnjYð_j;b^{vÞ1ðx;k;a;bÞ:} ð34Þ

Remark 5.2. Substitutinga¼

v

^{¼k¼1 andb¼einto}(34), we have Bnð1x;bÞ ¼ ð1Þⁿb¹Bnðx;b¹Þ

(cf.[33,30,34]). If we setb¼1 into the above equation, we have the following well-known result:

Bnð1xÞ ¼ ð1ÞⁿBnðxÞ (cf.[7,30]).

Acknowledgement

The authors would like to thank the referees for their valuable comments on this present paper.

This paper was supported by Akdeniz University Scientific Research Projects Unit. Research Projects of Uludag university project number UAP (F)-2012/16.

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