A note on multi Poly-Euler numbers and Bernoulli p olynomials 1

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1 General Mathematics Vol 20, No 2-3 (202, A note on multi Poly-Euler numbers and Bernoulli p olynomials Hassan Jolany, Mohsen Aliabadi, Roberto B Corcino, and MRDarafsheh Abstract In this paper we introduce the generalization of Multi Poly- Euler polynomials and we investigate some relationship involving Multi Poly-Euler polynomials Obtaining a closed formula for generalization of Multi Poly-Euler numbers therefore seems to be a natural and important problem 200 Mathematics Subject Classification: B73, A07 Key words and phrases: Euler numbers, Bernoulli numbers, Poly-Bernoulli numbers, Poly-Euler numbers, Multi Poly-Euler numbers and polynomials Introduction In the 7th century a topic of mathematical interst was finite sums of powers of integers such as the series (n or the series Received 08 Jun, 2009 Accepted for publication (in revised form 29 November,

2 A note on multi Poly-Euler numbers and Bernoulli polynomials (n 2 Theclosedformforthesefinitesumswereknown,but the sum of the more general series k +2 k ++(n k was notit was the mathematician Jacob Bernoulli who would solve this problembernoulli numbers arise in Taylor series in the expansion ( x = e x B n x n and we have, (2 S m (n = n k= km = m +2 m + +n m = m+ m k=0 ( m+ Bk n m+ k k and we have following matrix representation for Bernoulli numbers(for n N,[-4] (3 B n = ( n (n! n n n n 0 0 ( 3 2 ( n ( n 2 2 ( n (n n 2 n 2 Euler on page 499 of [5], introduced Euler polynomials, to evaluate the alternating sum (4 A n (m = m ( m k k n = m n (m n ++( m n k= The Euler numbers may be defined by the following generating functions (5 2 = e t + E n t n and we have following folowing matrix representation for Euler numbers, [,2,3,4]

3 24 Hassan Jolany, et al (6 E 2n = ( n (2n! 2! 4! 2! (2n 2! (2n! (2n 4! (2n 2! 2! 4! 2! The poly-bernoulli polynomials have been studied by many researchers in recent decade The history of these polynomials goes back to Kaneko The poly-bernoulli polynomials have wide-ranging application from number theory and combinatorics and other fields of applied mathematics One of applications of poly-bernoulli numbers that was investigated by Chad Brewbaker in [6,7,8,9], is about the number of (0, -matrices with n-rows and k columns He showed the number of (0,-matrices with n-rows and k columns uniquely reconstructable from their row and column sums are the poly-bernoulli numbers of negative index B n ( k Let us briefly recall poly-bernoulli numbers and polynomials For an integer k Z, put (7 Li k (z = z n n= n k which isthek-thpolylogarithmifk, andarational functionif k 0 The name of the function come from the fact that it may alternatively be defined as the repeated integral of itself The formal power series can be used to define Poly-Bernoulli numbers and polynomials The polynomials B (k n (x are said to be poly-bernoulli polynomials if they satisfy, (8 Li k ( e t e xt = B e t n (k (x tn In fact, Poly-Bernoulli polynomials are generalization of Bernoulli polynomials, because for n 0, we have,

4 A note on multi Poly-Euler numbers and Bernoulli polynomials 25 (9 ( n B ( n ( x = B n (x Sasaki,[0], Japanese mathematician, found the Euler type version of these polynomials, In fact, he by using the following relation for Euler numbers, (0 cosht = found a poly-euler version as follows ( Li k ( e 4t 4tcosht = E n t n E (k n tn Moreover, he by defining the following L-function, interpolated his definition about Poly-Euler numbers (2 L k (s = t s Li k ( e 4t dt Γ(s 0 4(e t +e t and Sasaki showed that (3 L k ( n = ( n n E(k n 2 But the fact is that working on such type of generating function for finding some identities is not so easy So by inspiration of the definitions of Euler numbers and Bernoulli numbers, we can define Poly-Euler numbers and polynomials as follows which also ABayad [], defined it by following method in same times Definition (Poly-Euler polynomials:the Poly-Euler polynomials may be defined by using the following generating function, (4 2Li k ( e t +e t e xt = E (k n tn

5 26 Hassan Jolany, et al Ifwereplacetby4tandtakex = /2andusing thedefinitioncosht = e t +e t 2, we get the Poly-Euler numbers which was introduced by Sasaki and Bayad and also we can find same interpolating function for them (with some additional constant coefficient The generalization of poly-logarithm is defined by the following infinite series (5 Li (k,k 2,,k r(z = m,m 2,,m r which here in summation (0 < m < m 2 < m r zmr m k mkr r Kim-Kim [2], one of student of Taekyun Kim introduced the Multi poly- Bernoulli numbers and proved that special values of certain zeta functions at non-positive integers can be described in terms of these numbers The study of Multi poly-bernoulli numbers and their combinatorial relations has received much attention in [6-3] The Multi Poly-Bernoulli numbers may be defined as follows (6 Li (k,k 2,,kr( e t ( e t r = B (k,k 2,,k r t n n So by inspiration of this definition we can define the Multi Poly-Euler numbers and polynomials Definition 2 Multi Poly-Euler polynomials E (k,,k r n (x, (n = 0,,2, are defined for each integer k,k 2,,k r by the generating series (7 2Li (k,,kr( e t e rxt = E (k,,k r (+e t r n (x tn andifx = 0, thenwecandefinemultipoly-eulernumberse (k,,k r n = E (k,,k r n (0 Now we define three parameters a, b, c, for Multi Poly-Euler polynomials and Multi Poly-Euler numbers as follows

6 A note on multi Poly-Euler numbers and Bernoulli polynomials 27 Definition 3 Multi Poly-Euler polynomialse (k,,k r n (x,a,b, (,,2, are defined for each integer k,k 2,,k r by the generating series (8 2Li (k,,kr( (ab t (a t +b t r e rxt = E (k,,k r n (x,a,b tn In the same way, and if x = 0, then we can define Multi Poly-Euler numbers with a,b parameters E (k,,k r n (a,b = E (k,,k r n (0,a,b In the following theorem, we find a relation between E (k,,k r n (a, b and E (k,,k r n (x Theorem Let a,b > 0, ab ± then we have (9 E (k,k 2,,k r n (a,b = E (k,k 2,,k r ( n lna+lnb (lna+lnbn Proof By applying the Definition 2 and Definition 3,we have 2Li (k,,k r( (ab t (a t +b t r = E (k,,k r n (a,b tn = e rtlna2li (k,,k r( e tlnab (+e tlnab r So, we get 2Li (k,,k r( (ab t (a t +b t r = ( E (k,,k r lna n (lna+lnb ntn lna+lnb Therefore, by comparing the coefficients of t n on both sides, we get the desired result Now, In next theorem, we show a shortest relationship between E (k,k 2,,k r n (a,b and E (k,k 2,,k r n

7 28 Hassan Jolany, et al Theorem 2 Let a,b > 0, ab ± then we have (20 E (k,k 2,,k r n (a,b = n i=0 Proof By applying the Definition 2, we have, E (k,,k r n (a,b tn = 2Li (k,,k r( (ab t (a t +b t r r n i (lna+lnb i (lna n i( n (k i E,k 2,,k r i = e rtlna2li (k,,k r( e tlnab (+e tlnab ( r ( r k t k (lna k = E (k,,k r n (lna+lnb ntn k! k=0 ( j,,k r = r j ie(k j (lna+lnb i (lna j i t j i!(j i! j=0 i=0 So, by comparing the coefficients of t n on both sides, we get the desired result By applying the definition 2, by simple manipulation, we get the following corollary Corollary For non-zero numbers a,b, with ab we have (2 E (k,,k r n (x;a,b = n ( n i r n i E (k,,k r i (a,bx n i i=0 Furthermore, by combinig the results of Theorem 2, and Corollary, we get the following relation between generalization of Multi Poly-Euler polynomials with a,b parameters E (k,,k r n (x; a, b, and Multi Poly-Euler numbers E (k,,k r n

8 A note on multi Poly-Euler numbers and Bernoulli polynomials 29 (22 E (k,,k r n (x;a,b = n k k=0j=0 r n k( n k k( j (lna k j (lna+lnb j E (k,,k r j x n k Now, we state the Addition formula for generalized Multi Poly- Euler polynomials Corollary 2 (Addition formula For non-zero numbers a, b, with ab we have (23 E (k,,k r n (x+y;a,b = n ( n k r n k E (k,,k r k (x;a,by n k k=0 Proof We can write E (k,,k r n (x+y;a,b tn = 2Li (k,,k r( (ab t e (x+yrt (a t +b t r = 2Li (k,,k r( (ab t e xrt e yrt (a t +b t ( r ( n = E (k,,k r n (x;a,b tn y i r i t i i! i=0 ( n ( n = r n k y n k E (k,,k r t n k (x;a,b k k=0 So, by comparing the coefficients of t n on both sides, we get the desired result 2 Explicit formula for Multi Poly-Euler polynomials Here we present an explicit formula for Multi Poly-Euler polynomials

9 30 Hassan Jolany, et al Theorem 3 The Multi Poly-Euler polynomials have the following explicit formula (24 E (k,k 2,,k r n (x= n i=0 0 m m 2 mr c +c 2 +=r m r j=0 2(rx j n i r!( j+c +2c 2 + (c +2c 2 +v i ( mr (c!c 2!(m k mk 2 2 mkr r j ( n i Proof We have Li (k,k 2,,k r( e t e rxt = 0 m m 2 m r ( e tmr m k mk 2 2 mkr r e rxt = 0 m m 2 m r m k m k 2 = ( On the other hand, 0 m m 2 m r j=0 2 m kr r m r m r j=0 ( ( j mr j t n ( j (rx j n( m r j m k mk 2 2 mkr r ( ( r r = ( n e nt +e t = r!( c +2c 2 + c!c 2! Hence, c +c 2 +=r e t(c +2c 2 + (rx j ntn = r!( c +2c 2 + (c +2c 2 + ntn c c +c 2 +=r!c 2! = ( r!( c +2c 2 + (c +2c 2 + n t n c c +c 2 +=r!c 2! 2Li (k,k 2,,k r( e t ( r e rxt = 2Li (+e t r (k,k 2,,k r( e t e rxt +e t

10 A note on multi Poly-Euler numbers and Bernoulli polynomials 3 = 2 = ( ( ( ( ( n i=0 m r 0 m m 2 m r j=0 c +c 2 +=r 0 m m 2 m r j=0 ( ( j (rx j n( m r j t n m k mk 2 2 mkr r r!( c +2c 2 + (c +2c 2 + n m r c +c 2 +=r c!c 2! ( j (rx j n i( m r j m k m k 2 2 m kr r t n t n i (n i! r!( c +2c 2 + (c +2c 2 + i c!c 2! t i i! =2 n i=0 0 m m 2 mr c +c 2 +=r m r j=0 (rx j n i r!( j+c +2c 2 + (c +2c 2 + i( m r ( n i (c!c 2!(m k mk 2 2 mkr r By comparing the coefficient of t n /, we obtain the desired explicit formula Definition 4 (Poly-Euler polynomials with a, b, c parameters:the Poly- Euler polynomials with a,b,c parameters may be defined by using the following generating function, 2Li (25 k ( (ab t c xt = E (k a t +b t n (x;a,b,c tn Now, in next theorem, we give an explicit formula for Poly-Euler polynomials with a, b, c parameters Theorem 4 The generalized Poly-Euler polynomials with a, b, c parameters have the following explicit formula j tn (26 n m j m=0 j=0i=0 E (k n (x;a,b,c = 2( m j+i j k ( j i (xlnc (m j+i+lna (m j+i+lnb n

11 32 Hassan Jolany, et al Proof We can write E (k n (x;a,b,c tn = 2Li k( (ab t a t ((ab t + cxt ( ( = 2a t ( n (ab nt ( (ab t m m k =a m j ( t 2( m j+i j (ab t(x+m j+i c xt j k i m 0 j=0 i=0 = m j 2( m j+i j k m 0 j=0 i=0 m j m 0 j=0 i=0 = n m j m=0j=0i=0 2( m j+i j k 2( m j+i j k c xt ( j e t(x+m j+iln(ab e tlna e xtlnc = i ( j (xlnc (m j+i+lna (m j+ilnb ntn i ( j i (xlnc (m j+i+lna (m j+ilnb ntn By comparing the coefficient of t n /, we obtain the desired explicit formula References [] T M Apostol, On the Lerch Zeta function, Pacific J Math no, 95, 6-67 [2] G Dattoli, S Lorenzutta and C Cesarano, Bernoulli numbers and polynomials from a more general point of view, Rend Mat Appl Vol 22, No7, 2002, [3] H Jolany, R E Alikelaye and S S Mohamad, Some results on the generalization of Bernoulli, Euler and Genocchi polynomials, Acta Universitatis Apulensis,No 27,20, pp

12 A note on multi Poly-Euler numbers and Bernoulli polynomials 33 [4] S Araci, M Acikgoz and E en, On the extended Kim s p-adic q- deformed fermionic integrals in the p-adic integer ring, Journal of Number Theory 33 ( [5] LEuler, Institutiones Calculi Differentialis, Petersberg,755 [6] C Brewbaker, Lonesum (0,-matrices and poly-bernoulli numbers of negative index, Masters thesis, Iowa State University, 2005 [7] M Kaneko, Poly-Bernoulli numbers, J Thorie de Nombres 9 ( [8] Y Hamahata and H Masubuchi, Recurrence formulae for multipoly-bernoulli numbers, Integers 7 (2007, A46 [9] Hassan Jolany, Explicit formula for generalization of Poly-Bernoulli numbers and polynomials with a,b,c parameters, arxiv:09387 [0] Y Ohno and Y Sasaki, On poly-euler numbers, preprint [] A Bayad, Y Hamahata, Poly-Euler polynomials and Arakawa- Kaneko type zeta functions, preprint [2] M-S Kim and T Kim, An explicit formula on the generalized Bernoulli number with order n, Indian J Pure Appl Math 3 (2000, [3] H Jolany, MR Darafsheh, RE Alikelaye, Generalizations of Poly- Bernoulli Numbers and Polynomials, Int J Math Comb 200, No 2, 7-4

13 34 Hassan Jolany, et al Hassan Jolany Universit des Sciences et Technologies de Lille UFR de Mathmatiques Laboratoire Paul Painlev CNRS-UMR Villeneuve d Ascq Cedex/France hassanjolany@mathuniv-lillefr Mohsen Aliabadi Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, USA mohsenmath88@gmailcom Roberto B Corcino Department of Mathematics Mindanao State University, Marawi City, 9700 Philippines rcorcino@yahoocom MRDarafsheh Department of Mathematics, Statistics and Computer Science Faculty of Science University of Tehran, Iran darafsheh@utacir

\displaystyle \frac{\mathrm{l}\mathrm{i}_{k}(1-e^{-4t})}{4t(\cosh t)}=\sum_{n=0}^{\infty}\frac{e_{n}^{(k)}}{n!}t^{n}

$\displaystyle \frac{\mathrm{l}\mathrm{i}_{k}(1-e^{-4t})}{4t(\cosh t)}=\sum_{n=0}^{\infty}\frac{e_{n}^{(k)}}{n!}t^{n}$ RIMS Kôkyûroku Bessatsu B32 (2012), 271 278 On the parity of poly Euler numbers By Yasuo \mathrm{o}\mathrm{h}\mathrm{n}\mathrm{o}^{*} and Yoshitaka SASAKI ** Abstract Poly Euler numbers are introduced