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Relationships between Mahler Expansion and Higher Order

q-Daehee polynomials

Ugur Duran1; and Mehmet Acikgoz2 1Department of the Basic Concepts of Engineering,

Faculty of Engineering and Natural Sciences, Iskenderun Technical University, TR-31200 Hatay, Turkey

E-Mail: mtdrnugur@gmail.com & ugur.duran@iste.edu.tr Corresponding Author

2University of Gaziantep, Faculty of Science and Arts, Department of Mathematics, TR-27310 Gaziantep, Turkey

E-Mail: acikgoz@gantep.edu.tr

Abstract

In this paper, we derive multifarious relationships among the two types of higher order q-Daehee polynomials and p-adic gamma function via Mahler theorem. Also, we compute some weightedp-adicq-integrals of the derivative of p-adic gamma function related to the Stirling numbers of the both kinds and theq-Bernoulli polynomials of orderk.

2010 Mathematics Subject Classi…cation. Primary 05A10, 05A30; Secondary 11B65, 11S80, 33B15.

Key Words and Phrases.q-numbers,p-adic numbers,p-adic gamma function, Mahler expansion, Bernoulli polynomials, Daehee polynomials, Stirling numbers of the …rst kind, Stirling numbers of the second kind.

1. Introduction

The p-adic numbers are a counterintuitive arithmetic system, which were primarily considered by the Kummer in circa 1850. With the consideration of these numbers, many mathematicians and physicists started to develop new scienti…c tools using their available, useful and applicable properties. Firstly, Kurt Hensel who is a German mathematician (1861-1941) advanced the p-adic numbers in a study associated with the improvement of algebraic numbers in power series in about1897. Several e¤ects of these researches have emerged in mathematics and physics such asp-adic analysis, string theory, p-adic quantum mechan-ics, quantum …eld theory, representation theory, algebraic geometry, complex systems, dynamical systems, genetic codes and so on (cf. [1-4;6-17]; also see the references cited in each of these earlier studies). The one important tool of the mentioned advancements is p-adic gamma function which is introduced by Yasou Morita [14] in1975. Intense research activities in such an area asp-adic gamma function is principally moti-vated by their signi…cance inp-adic analysis. Hereby, in recent fourty years,p-adic gamma function and its diverse extensions have been studied and progressed broadly by many mathematicians,cf. [4;6-8;13;14;16]; see also the related references cited therein.

The Daehee polynomials, denoted byDn(x), are de…ned by means of the following exponential generating function (cf. [10]):

1 X

n=0 Dn(x)t

n

n! =

log (1 +t) t (1 +t)

x

. (1.1)

In the casex= 0 in (1.1), one can getDn(0) :=Dn standing for n-th Daehee number, see [2;4;9;10;15;17] for more detailed information about these related issues.

Throughtout this paper, we shall use the following …gures: Zindicates the set of all integers,Qindicates the …eld of rational numbers,Zpindicates the ring of thep-adic integers,Qpindicates the …eld of thep-adic

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numbers, and Cp indicates the p-adic completion of an algebraic closure of Qp, where pbe a …xed prime number (cf. [1-4;6-17]). LetN=f1;2;3; gandN =N[ f0g.

The familiarp-adic Haar distribution 0and the Volkenborn integral (p-adic integral)I0(f)of a function f 2U D(Zp;Cp) =ffjf :Zp!Cp is uniformly di¤erentiable functiong;

respectively, are given by (cf. [2;4;6;7;9-12;15-17])

0 y+pNZp = 1=pN (1.2)

and

I0(f) = Z

Zp

f(x)d 0(x) = lim N!1

1 pN

pN 1

X

x=0

f(x) (1.3)

which yields the Daehee polynomialsDn(x)and Daehee numbersDn, forn2N , as follows Dn(x) =

Z

Zp

(x+y)nd 0(y) andDn= Z

Zp

(y)nd 0(y), where the symbol(x)n denotes the falling factorial given by (cf. [2;4;9;10;15;17])

(x)n =x(x 1) (x 2) (x n+ 1). (1.4)

The falling factorial(x)n satis…es the following identity: (x)n=

n X

k=0

S1(n; k)xk, (1.5)

whereS1(n; k)denotes the Stirling number of the …rst kind (see [2;4;9;10;15;17]).

The notationq may be varyingly considered as indeterminate, complex numberq2Cwith 0<jqj<1, or p-adic numberq 2Cp with jq 1jp < p

1

p 1 so that qx = exp (xlogq) forjxj

p 1, where j.jp indicates thep-adic norm on Cp normalized byjpjp= 1=p.

Theq-Volkenborn integral orp-adicq-integral onZpof a function f 2U D(Zp)is de…ned (cf. [2;4;6;9;13;15]) as follows:

Iq(f) = Z

Zp

f(x)d q(x) = lim N!1

1 [pN]

q pN 1

X

x=0

f(x)qx. (1.6)

Suppose thatf1(x) =f(x+ 1). Then, we see that

qIq(f1) =Iq(f) + (q 1)f(0) + q 1

logqf

0

(0): (1.7)

For these related issues, see [2;4;6;9;13;15] and related references cited therein.

The higher orderq-Daehee numbersDn;q(k) and higher orderq-Daehee polynomialsDn;q(k)(x)are de…ned (cf. [2]) by means ofq-Volkenborn integrals fork2N:

Dn;q(k) = Z

Zp

Z

Zp

(x1+ +xk)nd q(x1) d q(xk) (n 0); (1.8)

Dn;q(k)(x) = Z

Zp

Z

Zp

(x1+ +xk+x)nd q(x1) d q(xk) (n 0). (1.9)

Whenk= 1, it is obvious thatlimq!1D(1)n;q:=Dn andlimq!1D(1)n;q(x) :=Dn(x).

The higher orderq-Daehee numbers and polynomials of the second kind are introduced by the following q-Volkenborn integrals:

b D(k)n;q =

Z

Zp

Z

Zp

( x1 xk)nd q(x1) d q(xk) (n 0); (1.10) b

Dn;q(k)(x) = Z

Zp

Z

Zp

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Theq-Daehee polynomials and numbers and their various generalizations have been studied by many math-ematicians,cf. [2;4;6;9;13;15]; see also the related references cited therein.

In the present paper, we are interested in constructing the correlations between thep-adic gamma function and the higher orderq-Daehee polynomials and numbers by using the methods of theq-Volkenborn integral and Mahler series expansion; in the last part, we examine the results derived in this paper.

2. Higher Order q-Daehee Polynomials Associated with Mahler Theorem

This section provides some properties, identities and correlations for the mentioned gamma function, the higher order q-Daehee polynomials and numbers of the both types, Stirling numbers of the …rst kind and p-adic Euler constant by using the Mahler expansion of thep-adic gamma function given by Proposition 1 below.

Thep-adic gamma function is de…ned as follows p(x) = lim

n!x( 1)

n Y

j<n

(p;j)=1

j (x2Zp), (2.1)

wherenapproachesxthrough positive integers.

Thep-adic gamma function in conjunction with its several extensions andp-adic Euler constant have been developed by many physicists and mathematicians,cf. [4;6-8;13;14;16]; see also the references cited in each of these earlier works.

Forx2Zp, the symbol nx is given by xn = x(x 1)n!(x n+1) (n2N)with x0 = 1.

Letx2Zp andn2N. The functions x! xn form an orthonormal base of the spaceC(Zp!Cp)with respect to the Euclidean normk k1. The mentioned orthonormal base satis…es the following equality:

x n

0 =

nX1

j=0

( 1)n j 1 n j

x

j (see [6;7;13;16]). (2.2) Mahler investigated a generalization for continuous maps of ap-adic variable utilizing the special polynomials as binomial coe¢ cient polynomial [12] in1958. It implies that for anyf 2C(Zp!Cp), there exist unique elementsa0; a1; a2; : : :ofCp such that

f(x) = 1 X

n=0 an

x

n (x2Zp): (2.3)

The base n :n2N is named as Mahler base of the spaceC(Zp!Cp);and the componentsfan:n2Ng inf(x) =P1n=0an xn are called Mahler coe¢ cients off 2C(Zp!Cp). The Mahler expansion of thep-adic gamma function p and its Mahler coe¢ cients are determined in [16] as follows.

Proposition 1. For x 2 Zp, let p(x+ 1) = P1n=0an xn be Mahler series of p. Then its coe¢ cients

satisfy the following identity:

X

n=0

( 1)n+1an xn

n! = 1 xp

1 x exp x+ xp

p . (2.4)

We now provide the higher orderq-Volkenborn integral of thep-adic gamma function via the higher order q-Daehee polynomials by Theorem 1.

Theorem 1. Let x; xi2Zp wherei2 f1;2; : : : kg. We have Z

Zp

Z

Zp

p(x1+ +xk+x+ 1)d q(x1) d q(xk) = 1 X

n=0 an

D(k)n;q(x)

n! , (2.5)

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Proof. Forx; xi 2Zp wherei2 f1;2; : : : kg, by using the relation x1+xn 2 = (x1+xn!2)n and Proposition 1, we get

Z

Zp

Z

Zp

p(x1+ +xk+x+ 1)d q(x1) d q(xk)

= Z

Zp

Z

Zp

1 X

n=0 an

(x1+ +xk+x)n

n! d q(x1) d q(xk)

= 1 X

n=0 an

1 n!

Z

Zp

Z

Zp

(x1+ +xk+x)nd q(x1) d q(xk);

which is the desired result (2.5) via the formula (1.9).

We now examine a consequence of the Theorem 1 as follows.

Corollary 1. Choosingx= 0 in Theorem 1 gives the following relation for p andD(k)n;q: Z

Zp

Z

Zp

p(x1+ +xk+ 1)d q(x1) d q(xk) = 1 X

n=0 an

Dn;q(k)

n! , (2.6)

wherean is given by Proposition 1.

Letn2N0andk2N. Choet al. [2] gave the following correlation: D(k)n;q(x) =

n X

m=0

S1(n; m)B(k)m;q(x), (2.7)

whereB(k)m;q(x)denotes them-th Bernoulli polynomials of orderkde…ned by Bm;q(k)(x) =

Z

Zp

Z

Zp

(x1+ +xk+x)md q(x1) d q(xk) (m2N0). (2.8) As a result of Theorem 1 and relation (2.7), one other higher orderq-Volkenborn integrals of the p-adic gamma function via theq-Bernoulli polynomials of orderkis stated below.

Corollary 2. Forx; xi2Zp wherei2 f1;2; : : : kg, we acquire Z

Zp

Z

Zp

p(x1+ +xk+x+ 1)d q(x1) d q(xk) = 1 X

n=0 n X

m=0 an

S1(n; m) n! B

(k) m;q(x).

Here is thep-adicq-integrals of the derivative of the p-adic gamma function.

Theorem 2. Forx; xi 2Zp wherei2 f1;2; : : : kg, we have Z

Zp

Z

Zp

0

p(x1+ +xk+x+ 1)d q(x1) d q(xk) = 1 X

n=0 nX1

j=0 an

( 1)n j 1Dj;q(k)(x) (n j)j! .

Proof. In view of Proposition 1, we obtain Z

Zp

Z

Zp

0

p(x1+ +xk+x+ 1)d q(x1) d q(xk)

= Z

Zp

Z

Zp

1 X

n=0 an

x1+ +xk+x n

0

d q(x1) d q(xk)

= 1 X

n=0 an

Z

Zp

Z

Zp

x1+ +xk+x n

0

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and using (2.2), we derive Z

Zp

Z

Zp

0

p(x1+ +xk+x+ 1)d q(x1) d q(xk)

= 1 X

n=0 n 1 X

j=0 an

( 1)n j 1

n j

Z

Zp

Z

Zp

x1+ +xk+x

j d q(x1) d q(xk)

= 1 X

n=0 n 1 X

j=0 an

( 1)n j 1

n j

D(k)j;q(x) j! :

The immediate result of Theorem 2 is given as follows.

Corollary 3. Forxi2Zp where i2 f1;2; : : : kg;we have Z

Zp

Z

Zp

0

p(x1+ +xk+ 1)d q(x1) d q(xk) = 1 X

n=0 nX1

j=0 an

( 1)n j 1D(k)j;q (n j)j! .

We state the following theorem including a relation between p(x)andDbn;q(k)(x).

Theorem 3. Forx; xi 2Zp wherei2 f1;2; : : : kg, we have Z

Zp

Z

Zp

p( x1 xk x+ 1)d q(x1) d q(xk) = 1 X

n=0 an

b D(k)n;q(x)

n! ;

wherean is given by Proposition 1.

Proof. Forx; xi2Zp whereilies inf1;2; : : : kg, by utilizing the relation x1n x2 =

( x1 x2)n

n! and Proposi-tion 1, we get

Z

Zp

Z

Zp

p( x1 xk+x+ 1)d q(x1) d q(xk)

= Z

Zp

Z

Zp

1 X

n=0 an

( x1 xk+x)n

n! d q(x1) d q(xk)

= 1 X

n=0 an

1 n!

Z

Zp

Z

Zp

( x1 xk+x)nd q(x1) d q(xk);

which is the desired result with (1.11).

A consequence of Theorem 3 is given by the following corollary.

Corollary 4. Upon settingx= 0in Theorem 3 gives the following relation for p andDbn;q(k): Z

Zp

Z

Zp

p( x1 xk+ 1)d q(x1) d q(xk) = 1 X

n=0 an

b Dn;q(k)

n! ,

wherean is given by Proposition 1.

Letn2N0andk2N. Cho et al. [2] gave the following correlation: b

Dn;q(k)(x) = n X

m=0

( 1)n mS1(n; m)Bm;q(k) ( x),

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Corollary 5. Forx; xi2Zp wherei2 f1;2; : : : kg, we acquire Z

Zp

Z

Zp

p( x1 xk+x+ 1)d q(x1) d q(xk) = 1 X n=0 n X m=0

( 1)n man

S1(n; m) n! B

(k) m;q( x).

We observe that

( y)n = ( y) ( y+ 1) ( y n+ 1) = ( 1)ny(y 1) (y+n 1)

= ( 1)n n X

m=0

S2(n; m)ym;

where S2(n; m) denotes the Stirling numbers of the second kind, cf. [2;4;9;10;15;17], de…ned by the following exponential generating function

(et 1)m

m! =

X

n 0

S2(n; m) tn n!:

We lastly state one otherp-adicq-integrals of the derivative of thep-adic gamma function by means of theq-Bernoulli polynomials of orderkand the Stirling numbers of the second kind.

Theorem 4. Forx; xi 2Zp wherei2 f1;2; : : : kg, we have Z

Zp

Z

Zp

0

p( x1 xk x+ 1)d q(x1) d q(xk) = 1 X

n=0 nX1

j=0 j X

m=0

S2(j; m)an

( 1)n 1 n j

Bj;q(k)( x) j! .

Proof. Via Proposition 1, we get Z

Zp

Z

Zp

0

p( x1 xk+x+ 1)d q(x1) d q(xk)

= Z Zp Z Zp 1 X n=0 an

x1 xk+x n

0

d q(x1) d q(xk)

= 1 X n=0 an Z Zp Z Zp

x1 xk+x n

0

d q(x1) d q(xk)

and utilizing (2.2) and (2.8), we derive Z

Zp

Z

Zp

0

p( x1 xk+x+ 1)d q(x1) d q(xk)

= 1 X

n=0 nX1

j=0 an

( 1)n j 1

n j

Z

Zp

Z

Zp

x1 xk+x

j d q(x1) d q(xk)

= 1 X

n=0 nX1

j=0 an

( 1)n j 1 (n j)j!

Z

Zp

Z

Zp

( x1 xk+x)jd q(x1) d q(xk)

= 1 X

n=0 nX1

j=0 an

( 1)n j 1 (n j)j!

Z

Zp

Z

Zp

( 1)j j X

m=0

S2(j; m) (x1+ +xk x)md q(x1) d q(xk)

= 1 X

n=0 nX1

j=0 j X

m=0

S2(j; m)an

( 1)n 1 n j

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3. Conclusion and Observations

In the present paper, by means of the higher orderq-Daehee numbers and polynomials, we have discovered someq-Volkenborn integrals ofp-adic gamma function via their Mahler expansions. The calculations of the p-adic gamma function have been derived in terms of the higher orderq-Daehee polynomials and numbers and higher orderq-Daehee polynomials and numbers of the second kind. Moreover, we have investigated higher order q-Volkenborn integral of the derivative ofp-adic gamma function. The approach given in this paper is general and can be made by using the di¤erentp-adic integral and choosing the appropriate polynomials and numbers. Some results derived in this paper reduce to the results in the paper [4].

Competing Interests. The authors declare that they have no competing interests.

Authors´ Contributions. The authors declare that the work was realized in collaboration with the same responsibility. All authors read and approved the …nal manuscript.

References

[1] S. Araci, E. A¼gyüz, M. Acikgoz, On a q-analog of some numbers and polynomials, J. Inequal. Appl., 19, 2015 doi: 10.1186/s13660-014-0542-y.

[2] Y.K. Cho, T. Kim, T. Mansour, S.-H. Rim, Higher order q-Daehee polynomials, J. Comput. Anal. Appl., 19 (1), 2015, 167–173.

[3] J. Diamond, Thep-adic log gamma function andp-adic Euler constant, Trans. Amer. Math. Soc., 233, 1977, 321-337. [4] U. Duran, M. Acikgoz, Onp-adic gamma function related toq-Daehee polynomials and numbers, accepted for publication

in Proyecciones J. Math., 2018.

[5] B. N. Guo, F. Qi, Some identities and an explicit formula for Bernoulli and Stirling numbers, Anal. Appl., 19 (1), 2015, 167-173.

[6] Ö. Ç. Havara and H. Menken, On the Volkenborn integral of theq-extension of thep-adic gamma function, J. Math. Anal., 8 (2), 2017, 64-72.

[7] Ö. Ç. Havare and H. Menken, The Volkenborn integral of thep-adic gamma function, Int. J. Adv. Appl. Funct., 5 (2), 2018, 56-59.

[8] Y. S. Kim, q-analogues ofp-adic gamma functions andp-adic Euler constants, Comm. Korean Math. Soc., 13 (4), 1998, 735–741.

[9] T. Kim, S.-H. Lee, T. Mansour and J.-J. Seo, A note onq-Daehee polynomials and numbers, Adv. Stud. Contemp. Math., 24 (2), 2014, 155-160.

[10] D. S. Kim and T. Kim, Daehee numbers and polynomials, Appl. Math. Sci., 7 (120), 2013, 5969-5976. [11] N. Koblitz,p-adic numbers,p-adic analysis, and Zeta functions (Springer-Verlag, New York Inc, 1977).

[12] K. Mahler, An interpolation series for continuous functions of ap-adic variable, J. Reine Angew. Math., 199, 1958, 23-34. [13] H. Menken, A. Körükçü, Some properties of the q-extension of the p-adic gamma function, Abst. Appl. Anal., Volume

2013, Article ID 176470, 4 pages.

[14] Y. Morita, Ap-adic analogue of the -function, J. Fac. Sci., 22 (2), 1975, 255-266.

[15] H. Ozden, I. N. Cangul, Y. S¬msek, Remarks onq-Bernoulli numbers associated with Daehee numbers, Adv. Stud. Contemp. Math., 18 (1), 2009, 41-48.

[16] A. M. Robert, A course inp-adic analysis (Springer-Verlag New York, Inc., 2000).

References

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