International Journal of Analysis and Applications ISSN 2291-8639 Volume 14, Number 2 (2017), 140-146 http://www.etamaths.com IDENTITIES ON GENOCCHI POLYNOMIALS AND GENOCCHI NUMBERS CONCERNING BINOMIAL COEFFICIENTS QING ZOU∗ Abstract. In this paper, the author gives some new identities on Genocchi polynomials and Genoc- chi numbers. 1. Introduction The researches on Genocchi numbers and Genocchi polynomials have a long history. It can be traced back to Angelo Genocchi (1817–1889). Nowadays, Genocchi numbers and kinds of Genocchi polynomials have become a popular research topic. During these very recent years, some researchers such as Araci [1–7] did many researches on this interesting topic. They studied Genocchi numbers and Genocchi polynomials extensively in many branches of Mathematics, such as elementary number theory, analytic number theory, theory of modular forms, p-adic analytic number theory and etc.. Now, let us show the definitions of Genocchi numbers and Genocchi polynomials. The Genocchi numbers are a sequence of integers that satisfy the following exponential generating function 2t et + 1 = ∞∑ n=0 Gn tn n! , |t| < π, with the convention that replacing Gn by Gn. The first few Genocchi numbers are G0 = 0, G1 = 1, G2 = −1, G3 = 0, G4 = 1, G5 = 0, G6 = −3, G7 = 0, G8 = 17. The classic Genocchi polynomials are usually defined by mean of the following exponential gener- ating function 2t et + 1 ·ext = ∞∑ n=0 Gn(x) tn n! , |t| < π, with the convention that replacing Gn(x) by Gn(x). It is clear that Gn(0) = Gn. According to the classic Genocchi polynomials, some mathematicians introduced several new poly- nomials that extended the classic Genocchi polynomials. Araci [6] and Kim et al. [8] did some researches on the so-called Genocchi polynomials of order k, which were defined by ( 2t et + 1 )k ·ext = ∞∑ n=0 G(k)n (x) tn n! . Araci [6] and He [9, 10] introduced the Apostol–Genocchi polynomials defined by 2t λet + 1 ·ext = ∞∑ n=0 Gn(x,λ) tn n! . Received 9th March, 2017; accepted 25th May, 2017; published 3rd July, 2017. 2010 Mathematics Subject Classification. Primary 11B68; Secondary 05A19. Key words and phrases. Genocchi polynomial; Genocchi number; binomial inverse formula. c©2017 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 140 IDENTITIES ON GENOCCHI POLYNOMIALS AND GENOCCHI NUMBERS 141 Based on which, Araci [7] introduced the high order Apostol–Genocchi polynomials which can be called the generalized Apostol–Genocchi polynomials of order k ∈ C,( 2t λet + 1 )k ·ext = ∞∑ n=0 G(k)n (x,λ) tn n! . In [11], Lim defined the degenerated Genocchi polynomials G(k)n (x,λ) of order k to be( 2t (1 + λt)1/λ + 1 )k (1 + λt)x/λ = ∞∑ n=0 G(k)n (x,λ) tn n! . Besides these generalizations, Araci [1], Duran et al. [12] and Agyuz et al. [13] also introduced the q-alanogue of the Genocchi polynomials as follows, ∞∑ n=0 Gn,q(x) tn n! = t ∫ Zp q−yet[y+x]qdµ−q(y), where [x]q = 1 −qx 1 −q , [x]−q = 1 − (−q)x 1 + q . This definition used p-adic fermionic q-integral on Zp with respect to µ−q. It can also be defined by ∞∑ n=0 Gn,q(x) tn n! = [2]qt ∞∑ m=0 (−1)met[m+x]q. In which when we take x = 0, it becomes Gn,q(0) := Gn,q, which we call it the n-th q-Genocchi number. When it comes to Genocchi numbers, the most common thing comes to our mind is to research the relations between Genocchi numbers, Bernoulli numbers [14–16] and Euler numbers [14, 17]. Indeed, most researches on Genocchi numbers concern the relations between these three kinds of numbers (see for example [2–4, 18, 19]). In other words, there are many literatures that provide identities on these three kinds of numbers. Similarly, when it comes to Genocchi polynomials, the most common thing is to research on the relations between Genocchi polynomials, Bernoulli polynomials and Euler polynomials (see for example [2–4, 9, 18–21]). Even though when it comes to the generalized Genocchi numbers and generalized Genocchi polynomials, it is unavoidable to research the relations as above. In this paper, we do not want to find relations between the three kinds of numbers or the three kinds of polynomials. We will focus only on Genocchi numbers themselves and Genocchi polynomials themselves. In other words, in this paper, we will give some identities only concern Genocchi num- bers and Genocchi polynomials. Actually, by these identities combining with the identities between Genocchi numbers (polynomials), Bernoulli numbers (polynomials) and Euler numbers (polynomials), one can obtain some other identities. While we do not want to show them here since the process of combining two identities is not very novel. 2. Identities on Genocchi numbers and Genocchi polynomials Let us start this section with some straightforward derived identities on Genocchi numbers and Genocchi polynomials. Differentiating both sides of the exponential generating function for Gn with respect to x yields d dx Gn(x) = nGn−1(x), deg Gn+1(x) = n. By which we can get ∫ b a Gn(x)dx = Gn+1(b) −Gn+1(a) n + 1 . Thanks to [3, 18], we have Gn(x) = n∑ k=0 ( n k ) Gkx n−k. 142 Q. ZOU Combining the above two identities and the relation (2.7) below shows∫ 1 0 Gn(x)dx = { 0 n = 0 −2Gn+1 n+1 n ≥ 1 . Besides these classical identities, one can also find more identities concerning Genocchi numbers and Genocchi polynomials in [4]. Next, we show some new identities on Genocchi numbers and Genocchi polynomials. Theorem 2.1. For n ≥ 2, we have 1 2 n∑ k=0 ( n k ) Gk(x) ·Gn−k+1(1) n−k + 1 = x ·Gn(x) − n n + 1 Gn+1(x). (2.1) 1 2 n∑ k=0 ( n k ) Gk+1(x) ·Gn−k(1) k + 1 = x ·Gn(x) − n n + 1 Gn+1(x). (2.2) Proof. Let us recall the generating function of Genocchi polynomials first, 2t et + 1 ·ext = ∞∑ n=0 Gn(x) tn n! . Taking the partial derivative with respect to t on the right hand side, we deduce that ∂ ∂t ∞∑ n=0 Gn(x) tn n! = ∂ ∂t ( G0(x) + G1(x)t + G2(x) t2 2! + G3(x) t3 3! + · · · ) =G1(x) + G2(x)t + G3(x) t2 2! + · · · = ∞∑ n=0 Gn+1(x) tn n! . (2.3) Now, let us look at the left hand side. ∂ ∂t ( 2t et + 1 ·ext ) = (2ext + xext · 2t)(et + 1) − (2text ·et) (et + 1)2 = 1 t 2t ·ext et + 1 + x 2t ·ext et + 1 − 1 2t 2t ·ext et + 1 2t ·et et + 1 = 1 t ∞∑ n=0 Gn(x) tn n! + x ∞∑ n=0 Gn(x) tn n! − 1 2t ∞∑ n=0 Gn(x) tn n! ∞∑ n=0 Gn(1) tn n! = ∞∑ n=0 Gn+1(x) n + 1 tn n! + ∞∑ n=0 xGn(x) tn n! − 1 2 ∞∑ n=0 Gn(x) tn n! ∞∑ n=0 Gn+1(1) n + 1 tn n! = ∞∑ n=0 [ Gn+1(x) n + 1 + x ·Gn(x) − 1 2 n∑ k=0 ( n k ) Gk(x) Gn−k+1(1) n−k + 1 ] tn n! . (2.4) In the second to last step, we used the fact that G0(x) = 0. Comparing the coefficients of t n n! in (2.3) and (2.4) yields Gn+1(x) n + 1 + x ·Gn(x) − 1 2 n∑ k=0 ( n k ) Gk(x) Gn−k+1(1) n−k + 1 = Gn+1(x). Then (2.1) follows from rearranging the terms in this identity. Note that the second to last step can also be written as ∂ ∂t ( 2t et + 1 ·ext ) = ∞∑ n=0 Gn+1(x) n + 1 tn n! + ∞∑ n=0 xGn(x) tn n! − 1 2 ∞∑ n=0 Gn+1(x) n + 1 tn n! ∞∑ n=0 Gn(1) tn n! , IDENTITIES ON GENOCCHI POLYNOMIALS AND GENOCCHI NUMBERS 143 which gives us Gn+1(x) n + 1 + x ·Gn(x) − 1 2 n∑ k=0 ( n k ) Gk+1(x) k + 1 Gn−k(1) = Gn+1(x). Rearranging the terms above yeilds (2.2). This completes the proof. � Remark 2.1. According to the process of the proof above, one can also obtain that 1 2 n∑ k=0 ( n k ) Gn−k(x) ·Gk+1(1) k + 1 = x ·Gn(x) − n n + 1 Gn+1(x), (2.5) and 1 2 n∑ k=0 ( n k ) Gn−k+1(x) ·Gk(1) n−k + 1 = x ·Gn(x) − n n + 1 Gn+1(x). (2.6) But we should notice that (2.5) and (2.6) are equivalent to (2.1) and (2.2), respectively. This is because when k goes from 0 to n, n−k also goes from 0 to k. Hence if we replace k by n−k in (2.1) and (2.2), we can then get (2.5) and (2.6) respectively. From this point of view, we do not regard (2.5) and (2.6) as new identities. Corollary 2.1. For n ≥ 2, we have 1 2 n∑ k=0 ( n k ) Gk ·Gn−k+1(1) n−k + 1 = − n n + 1 Gn+1. 1 2 n∑ k=0 ( n k ) Gk+1 ·Gn−k(1) k + 1 = − n n + 1 Gn+1. Proof. This lemma follows from taking x = 0 in Theorem 2.1. � Having developed to this point, it is necessary to say something about Gn(1). Since 2t et + 1 ·et = ∞∑ n=0 Gn(1) tn n! . Then ∞∑ n=0 (Gn + Gn(1)) tn n! = ∞∑ n=0 Gn tn n! + ∞∑ n=0 Gn(1) tn n! = 2t. Thus, G1 + G1(1) = 2 and for n ≥ 2, Gn + Gn(1) = 0, which means Gn(1) = { 1, n = 1, −Gn, n ≥ 2. (2.7) So, in this sense, we can call the integer sequence Gn(1) the negative Genocchi numbers. With this fact, we can obtain Corollary 2.2. For n ≥ 2, we have 1 2 n−1∑ k=0 ( n k ) Gk(x) ·Gn−k+1 n−k + 1 = ( 1 2 −x) ·Gn(x) + n n + 1 Gn+1(x). (2.8) 1 2 n−2∑ k=0 ( n k ) Gk+1(x) ·Gn−k k + 1 = ( 1 2 −x) ·Gn(x) + n n + 1 Gn+1(x). (2.9) Proof. Since Gn(1) = −Gn except for n = 1. Then we can replace Gn−k+1(1) by −Gn−k+1 except for k = n. This gives us 1 2 Gn(x) − 1 2 n−1∑ k=0 ( n k ) Gk(x) ·Gn−k+1 n−k + 1 = x ·Gn(x) − n n + 1 Gn+1(x), which means (2.8) holds true. 144 Q. ZOU Similarly, we can show (2.9) through (2.2). � If we take x = 0 in Corollary 2.2, we can arrive at the following conclusion. Corollary 2.3. For n ≥ 2, we have 1 2 n−1∑ k=0 ( n k ) Gk ·Gn−k+1 n−k + 1 = 1 2 Gn + n n + 1 Gn+1. 1 2 n−2∑ k=0 ( n k ) Gk+1 ·Gn−k k + 1 = 1 2 Gn + n n + 1 Gn+1. Next, let us talk about Gn(x + y) which is given by 2t et + 1 ·e(x+y)t = ∞∑ n=0 Gn(x + y) tn n! . As the basic properties we have mentioned for Gn(x), Gn(x + y) has the same properties, such as∫ d c ∫ b a Gn(x + y)dxdy = Gn+2(a + c) −Gn+2(b + c) (n + 2)(n + 1) − Gn+2(a + d) −Gn+2(b + d) (n + 2)(n + 1) . Now, we would like to show some identities on Gn(x + y). Theorem 2.2. For y 6= 0, Gn(x + y) = n∑ k=0 ( n k ) Gk(x)y n−k. (2.10) Conversely, we have Gn(x) = (−1)n n∑ k=0 (−1)k ( n k ) Gk(x + y)y n−k. (2.11) Symmetrically, when x 6= 0, we have Gn(x + y) = n∑ k=0 ( n k ) Gk(y)x n−k, (2.12) and Gn(y) = (−1)n n∑ k=0 (−1)k ( n k ) Gk(x + y)x n−k. (2.13) Proof. Since ∞∑ n=0 Gn(x + y) tn n! = 2t et + 1 e(x+y)t = 2t et + 1 ext ·eyt = ∞∑ n=0 Gn(x) tn n! ∞∑ n=0 yn tn n! = ∞∑ n=0 ( n∑ k=0 ( n k ) Gn−k(x)y n−k ) tn n! . Comparing the coefficients of t n n! shows (2.10) holds true. The binomial inverse formula [22, pp.192, (5.48)] reads as an = n∑ k=0 ( n k ) (−1)kbk ⇔ bn = n∑ k=0 ( n k ) (−1)kak. Equation (2.10) can be rewritten as Gn(x + y) yn = n∑ k=0 ( n k ) Gk(x) yk . IDENTITIES ON GENOCCHI POLYNOMIALS AND GENOCCHI NUMBERS 145 Taking ak = Gk(x+y) yk and bk = (−1)k Gk(x) yk gives us (−1)n Gn(x) yn = n∑ k=0 (−1)k ( n k ) Gk(x + y) yk , which shows (2.11) holds. Since x and y are symmetric, then we can obtain (2.12) and (2.13) by changing the position of x and y. � Remark 2.2. If we want (2.11) and (2.13) to be more beautiful, we can replace k by n−k. Then we can have Gn(x) = n∑ k=0 (−1)k ( n k ) Gn−k(x + y)y k, and Gn(y) = n∑ k=0 (−1)k ( n k ) Gn−k(x + y)x k. Corollary 2.4. n∑ k=0 (−1)k+1 ( n k ) Gk = (−1)nGn + 2n. Proof. Thanks to [3, 18], we have Gn(x + 1) + Gn(x) n = 2xn−1. (2.14) Taking x = −1 in (2.14) shows Gn + Gn(−1) = 2n · (−1)n−1. (2.15) Let x = 0 and y = −1 in (2.10), we deduce that Gn(−1) = n∑ k=0 (−1)n−k ( n k ) Gk. (2.16) Plugging (2.16) in (2.15) shows Gn + n∑ k=0 (−1)n−k ( n k ) Gk = 2n · (−1)n−1. Then multiplying (−1)n−1 on the both sides proves this corollary. � Corollary 2.5. For n ≥ 2, we have Gn + n∑ k=0 ( n k ) Gk = 0. Proof. Let x = 1 and y = 0 in (2.12), we can get that Gn(1) = n∑ k=0 ( n k ) Gk. Since we have mentioned above that Gn(1) + Gn = 0 when n ≥ 2. Then the conclusion follows. � 146 Q. ZOU References [1] S. 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Kim, Some identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials, Adv. Stud. Contemp. Math. 20 (1) (2010), 23–28. [20] Y. He and T. Kim, General convolution identities for Apostol-Bernoulli, Euler and Genocchi polynomials, J. Nonlinear Sci. Appl. 9 (2016), 4780–4797. [21] T. Agoh, Convolution identities for Bernoulli and Genocchi polynomials, Electronic J. Combin. 21 (2014), Article ID P1.65. [22] R. L. Graham. D. E. Knuth and O. Patashnik, Concrete mathematics — a foundation for computer science, 2nd edn. Addison-Wesley Publishing Company, Reading, 1994. Department of Mathematics, The University of Iowa, Iowa City, IA 52242-1419, USA ∗Corresponding author: zou-qing@uiowa.edu 1. Introduction 2. Identities on Genocchi numbers and Genocchi polynomials References