On the 𝑞-Euler Numbers and Polynomials with Weight 0 <meta name="citation_title" content="On the q -Euler Numbers and Polynomials with Weight 0 " /> //// Hindawi Publishing Corporation Home Journals About Us About this Journal Submit a Manuscript Table of Contents Journal Menu Abstracting and Indexing Aims and Scope Annual Issues Article Processing Charges Articles in Press Author Guidelines Bibliographic Information Contact Information Editorial Board Editorial Workflow Free eTOC Alerts Reviewers Acknowledgment Subscription Information Open Special Issues Published Special Issues Special Issue Guidelines Abstract Full-Text PDF Full-Text HTML Full-Text ePUB Linked References How to Cite this Article Abstract and Applied Analysis Volume 2012 (2012), Article ID 795304, 7 pages doi:10.1155/2012/795304 Research Article On the 𝑞 -Euler Numbers and Polynomials with Weight 0 T. Kim and J. Choi Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea Received 13 October 2011; Accepted 29 November 2011 Academic Editor: Ibrahim Sadek Copyright © 2012 T. Kim and J. Choi. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The purpose of this paper is to investigate some properties of 𝑞 -Euler numbers and polynomials with weight 0. From those 𝑞 -Euler numbers with weight 0, we derive some identities on the 𝑞 -Euler numbers and polynomials with weight 0. 1. Introduction Let 𝑝 be a fixed odd prime number. Throughout this paper ℤ 𝑝 , ℚ 𝑝 , and ℂ 𝑝 will denote the ring of 𝑝 -adic rational integers, the field of 𝑝 -adic rational numbers, and the completion of algebraic closure of ℚ 𝑝 . The 𝑝 -adic absolute value is defined by | 𝑥 | 𝑝 = 1 / 𝑝 𝑟 where 𝑥 = 𝑝 𝑟 𝑠 / 𝑡 for 𝑠 , 𝑡 ∈ ℤ with ( 𝑝 , 𝑡 ) = ( 𝑝 , 𝑠 ) = 1 and 𝑟 ∈ ℚ . In this paper, we assume that 𝛼 ∈ ℚ and 𝑞 ∈ ℂ 𝑝 with | 1 − 𝑞 | 𝑝 < 1 . As well-known definition, the Euler polynomials are defined by 2 𝑒 𝑡 𝑒 + 1 𝑥 𝑡 = 𝑒 𝐸 ( 𝑥 ) 𝑡 = ∞  𝑛 = 0 𝐸 𝑛 ( 𝑡 𝑥 ) 𝑛 , 𝑛 ! ( 1 . 1 ) with the usual convention about replacing 𝐸 𝑛 ( 𝑥 ) by 𝐸 𝑛 ( 𝑥 ) (see [ 1 – 15 ]). In this special case, 𝑥 = 0 , 𝐸 𝑛 ( 0 ) = 𝐸 𝑛 are called the 𝑛 th Euler numbers (see [ 1 ]). Recently, the 𝑞 -Euler numbers with weight 𝛼 are defined by  𝐸 ( 𝛼 ) 0 , 𝑞  𝑞 = 1 , 𝑞 𝛼  𝐸 𝑞 ( 𝛼 )  + 1 𝑛 +  𝐸 ( 𝛼 ) 𝑛 , 𝑞 = 0 i f 𝑛 > 0 , ( 1 . 2 ) with the usual convention about replacing (  𝐸 𝑞 ( 𝛼 ) ) 𝑛 by  𝐸 ( 𝛼 ) 𝑛 , 𝑞 (see [ 3 , 12 ]). The 𝑞 -number of 𝑥 is defined by [ 𝑥 ] 𝑞 = ( 1 − 𝑞 𝑥 ) / ( 1 − 𝑞 ) (see [ 1 – 15 ]). Note that l i m 𝑞 → 1 [ 𝑥 ] 𝑞 = 𝑥 . Let us define the notation of 𝑞 -Euler numbers with weight 0 as  𝐸 ( 0 ) 𝑛 , 𝑞 =  𝐸 𝑛 , 𝑞 . The purpose of this paper is to investigate some interesting identities on the 𝑞 -Euler numbers with weight 0. 2. On the Extended 𝑞 -Euler Numbers of Higher-Order with Weight 0 Let 𝐶 ( ℤ 𝑝 ) be the space of continuous functions on ℤ 𝑝 . For 𝑓 ∈ 𝐶 ( ℤ 𝑝 ) , the fermionic 𝑝 -adic 𝑞 -integral on ℤ 𝑝 is defined by Kim as follows: 𝐼 𝑞 (  𝑓 ) = ℤ 𝑝 𝑓 ( 𝑥 ) 𝑑 𝜇 − 𝑞 ( 𝑥 ) = l i m 𝑁 → ∞ [ 2 ] 𝑞 1 + 𝑞 𝑝 𝑁 𝑝 𝑁 − 1  𝑥 = 0 𝑓 ( 𝑥 ) ( − 𝑞 ) 𝑥 , ( 2 . 1 ) (see [ 1 – 12 ]). By ( 2.1 ), we get 𝑞 𝑛 𝐼 𝑞  𝑓 𝑛  + ( − 1 ) 𝑛 − 1 𝐼 𝑞 [ 2 ] ( 𝑓 ) = 𝑞 𝑛 − 1  𝑙 = 0 ( − 1 ) 𝑛 − 1 − 𝑙 𝑓 ( 𝑙 ) 𝑞 𝑙 , ( 2 . 2 ) where 𝑓 𝑛 ( 𝑥 ) = 𝑓 ( 𝑥 + 𝑛 ) and 𝑛 ∈ ℕ (see [ 4 , 5 ]). By ( 1.2 ), ( 2.1 ), and ( 2.2 ), we see that  ℤ 𝑝 [ 𝑥 ] 𝑛 𝑞 𝛼 𝑑 𝜇 − 𝑞  𝐸 ( 𝑥 ) = ( 𝛼 ) 𝑛 , 𝑞 = [ 2 ] 𝑞 ( 1 − 𝑞 ) 𝑛 [ 𝛼 ] 𝑛 𝑞 𝑛  𝑙 = 0 ⎛ ⎜ ⎜ ⎝ 𝑛 𝑙 ⎞ ⎟ ⎟ ⎠ ( − 1 ) 𝑙 1 1 + 𝑞 𝛼 𝑙 + 1 . ( 2 . 3 ) In the special case, 𝑛 = 1 , we get  ℤ 𝑝 𝑒 𝑥 𝑡 𝑑 𝜇 − 𝑞 ( [ 2 ] 𝑥 ) = 𝑞 𝑞 𝑒 𝑡 = + 1 1 + 𝑞 − 1 𝑒 𝑡 + 𝑞 − 1 = ∞  𝑛 = 0 𝐻 𝑛  − 𝑞 − 1  𝑡 𝑛 , 𝑛 ! ( 2 . 4 ) where 𝐻 𝑛 ( − 𝑞 − 1 ) are the 𝑛 th Frobenius-Euler numbers. From ( 2.4 ), we note that the 𝑞 -Euler numbers with weight 0 are given by  𝐸 𝑛 , 𝑞 =  ℤ 𝑝 𝑥 𝑛 𝑑 𝜇 − 𝑞 ( 𝑥 ) = 𝐻 𝑛  − 𝑞 − 1  , f o r 𝑛 ∈ ℤ + . ( 2 . 5 ) Therefore, by ( 2.5 ), we obtain the following theorem. Theorem 2.1. For 𝑛 ∈ ℤ + , one has  𝐸 𝑛 , 𝑞 = 𝐻 𝑛  − 𝑞 − 1  , ( 2 . 6 ) where 𝐻 𝑛 ( − 𝑞 − 1 ) are called the 𝑛 th Frobenius-Euler numbers. Let us define the generating function of the 𝑞 -Euler numbers with weight 0 as follows:  𝐹 𝑞 ( 𝑡 ) = ∞  𝑛 = 0  𝐸 𝑛 , 𝑞 𝑡 𝑛 . 𝑛 ! ( 2 . 7 ) Then, by ( 2.3 ) and ( 2.7 ), we get  𝐹 𝑞 ( [ 2 ] 𝑡 ) = 𝑞 ∞  𝑚 = 0 ( − 1 ) 𝑚 𝑞 𝑚 𝑒 𝑚 𝑡 = 1 + 𝑞 𝑞 𝑒 𝑡 . + 1 ( 2 . 8 ) Now we define the 𝑞 -Euler polynomials with weight 0 as follows: ∞  𝑛 = 0  𝐸 𝑛 , 𝑞 ( 𝑡 𝑥 ) 𝑛 = 𝑛 ! 1 + 𝑞 𝑞 𝑒 𝑡 𝑒 + 1 𝑥 𝑡 . ( 2 . 9 ) Thus, ( 2.4 ) and ( 2.9 ), we get  ℤ 𝑝 𝑒 ( 𝑥 + 𝑦 ) 𝑡 𝑑 𝜇 − 𝑞 ( 𝑦 ) = 1 + 𝑞 𝑞 𝑒 𝑡 𝑒 + 1 𝑥 𝑡 = ∞  𝑛 = 0  𝐸 𝑛 , 𝑞 ( 𝑡 𝑥 ) 𝑛 . 𝑛 ! ( 2 . 1 0 ) From ( 2.10 ), we have ∞  𝑛 = 0  𝐸 𝑛 , 𝑞 ( 𝑡 𝑥 ) 𝑛 =  𝑛 ! 1 + 𝑞 − 1 𝑒 𝑡 + 𝑞 − 1  𝑒 𝑥 𝑡 = ∞  𝑛 = 0 𝐻 𝑛  − 𝑞 − 1  𝑡 , 𝑥 𝑛 , 𝑛 ! ( 2 . 1 1 ) where 𝐻 𝑛 ( − 𝑞 − 1 , 𝑥 ) are called the 𝑛 th Frobenius-Euler polynomials (see [ 9 ]). Therefore, by ( 2.11 ), we obtain the following theorem. Theorem 2.2. For 𝑛 ∈ ℤ + , one has  𝐸 𝑛 , 𝑞 (  𝑥 ) = ℤ 𝑝 ( 𝑥 + 𝑦 ) 𝑛 𝑑 𝜇 − 𝑞 ( 𝑥 ) = 𝐻 𝑛  − 𝑞 − 1  , , 𝑥 ( 2 . 1 2 ) where 𝐻 𝑛 ( − 𝑞 − 1 , 𝑥 ) are called the 𝑛 th Frobenius-Euler polynomials. From ( 2.2 ) and Theorem 2.2 , we note that 𝑞 𝑛 𝐻 𝑚  − 𝑞 − 1  , 𝑛 + 𝐻 𝑚  − 𝑞 − 1  = [ 2 ] 𝑞 𝑛 − 1  𝑙 = 0 ( − 1 ) 𝑙 𝑙 𝑚 𝑞 𝑙 , ( 2 . 1 3 ) where 𝑛 ∈ ℕ with 𝑛 ≡ 1 (mod 2). Therefore, by ( 2.13 ), we obtain the following corollary. Corollary 2.3. For 𝑛 ∈ ℕ , with 𝑛 ≡ 1 (mod 2) and 𝑚 ∈ ℤ + , one has 𝑞 𝑛 𝐻 𝑚  − 𝑞 − 1  , 𝑛 + 𝐻 𝑚  − 𝑞 − 1  = [ 2 ] 𝑞 𝑛 − 1  𝑙 = 0 ( − 1 ) 𝑙 𝑙 𝑚 𝑞 𝑙 . ( 2 . 1 4 ) In particular, 𝑞 = 1 , we get 𝐸 𝑚 ( 𝑛 ) + 𝐸 𝑚 ∑ = 2 𝑛 − 1 𝑙 = 0 ( − 1 ) 𝑙 𝑙 𝑚 , where 𝐸 𝑚 and 𝐸 𝑚 ( 𝑛 ) are called the 𝑚 th Euler numbers and polynomials which are defined by 2 𝑒 𝑡 = + 1 ∞  𝑚 = 0 𝐸 𝑚 𝑡 𝑚 , 2 𝑚 ! 𝑒 𝑡 𝑒 + 1 𝑥 𝑡 = ∞  𝑚 = 0 𝐸 𝑚 ( 𝑡 𝑥 ) 𝑚 . 𝑚 ! ( 2 . 1 5 ) By ( 2.2 ), we easily see that 𝑞  ℤ 𝑝 𝑓 ( 𝑥 + 1 ) 𝑑 𝜇 − 𝑞 (  𝑥 ) + ℤ 𝑝 𝑓 ( 𝑥 ) 𝑑 𝜇 − 𝑞 ( [ 2 ] 𝑥 ) = 𝑞 𝑓 ( 0 ) . ( 2 . 1 6 ) Thus, by ( 2.16 ), we get [ 2 ] 𝑞  = 𝑞 ℤ 𝑝 𝑒 ( 𝑥 + 1 ) 𝑡 𝑑 𝜇 − 𝑞 (  𝑥 ) + ℤ 𝑝 𝑒 𝑥 𝑡 𝑑 𝜇 − 𝑞 ( = 𝑥 ) ∞  𝑛 = 0  𝑞  ℤ 𝑝 ( 𝑥 + 1 ) 𝑛 𝑑 𝜇 − 𝑞  ( 𝑥 ) + ℤ 𝑝 𝑥 𝑛 𝑑 𝜇 − 𝑞  𝑡 ( 𝑥 ) 𝑛 = 𝑛 ! ∞  𝑛 = 0  𝑞 𝐻 𝑛  − 𝑞 − 1  , 1 + 𝐻 𝑛  − 𝑞 − 1 𝑡   𝑛 . 𝑛 ! ( 2 . 1 7 ) Therefore, by ( 2.16 ), we obtain the following theorem. Theorem 2.4. For 𝑛 ∈ ℤ + , one has 𝑞 𝐻 𝑛  − 𝑞 − 1  , 1 + 𝐻 𝑛  − 𝑞 − 1  =  1 + 𝑞 , i f 𝑛 = 0 , 0 , i f 𝑛 > 0 , ( 2 . 1 8 ) where 𝐻 𝑛 ( − 𝑞 − 1 , 𝑥 ) are called the 𝑛 th Frobenius-Euler polynomials and 𝐻 𝑛 ( − 𝑞 − 1 ) are called the 𝑛 th Frobenius-Euler numbers. In particular, 𝑞 = 1 , we have 𝐸 𝑛 ( 1 ) + 𝐸 𝑛 =  2 , i f 𝑛 = 0 , 0 , i f 𝑛 > 0 , ( 2 . 1 9 ) where 𝐸 𝑛 are called the 𝑛 th Euler numbers. From ( 2.5 ) and Theorem 2.2 , we note that  𝐸 𝑛 , 𝑞  ( 𝑥 ) = ℤ 𝑝 ( 𝑥 + 𝑦 ) 𝑛 𝑑 𝜇 − 𝑞 ( 𝑦 ) = 𝑛  𝑙 = 0 ⎛ ⎜ ⎜ ⎝ 𝑛 𝑙 ⎞ ⎟ ⎟ ⎠  ℤ 𝑝 𝑦 𝑙 𝑑 𝜇 − 𝑞 ( 𝑦 ) 𝑥 𝑛 − 𝑙 = 𝑛  𝑙 = 0 ⎛ ⎜ ⎜ ⎝ 𝑛 𝑙 ⎞ ⎟ ⎟ ⎠  𝐸 𝑛 , 𝑞 𝑥 𝑛 − 𝑙 =   𝐸 𝑥 + 𝑞  𝑛 , ( 2 . 2 0 ) where the usual convention about replacing (  𝐸 𝑞 ) 𝑙 by  𝐸 𝑙 , 𝑞 . By Theorems 2.2 and 2.4 , we get 𝑞  𝐸 𝑛 , 𝑞  𝐸 ( 1 ) + 𝑛 , 𝑞 =  [ 2 ] 𝑞 , i f 𝑛 = 0 , 0 , i f 𝑛 > 0 . ( 2 . 2 1 ) From ( 2.20 ) and ( 2.21 ), we have 𝑞   𝐸 𝑞  + 1 𝑛 +  𝐸 𝑛 , 𝑞 =  [ 2 ] 𝑞 , i f 𝑛 = 0 , 0 , i f 𝑛 > 0 . ( 2 . 2 2 ) For 𝑛 ∈ ℕ , by ( 2.20 ) and ( 2.22 ), we have 𝑞 2  𝐸 𝑛 , 𝑞 ( 2 ) = 𝑞 2   𝐸 𝑞  + 1 + 1 𝑛 = 𝑞 2 𝑛  𝑙 = 1 ⎛ ⎜ ⎜ ⎝ 𝑛 𝑙 ⎞ ⎟ ⎟ ⎠   𝐸 𝑞  + 1 𝑙   𝐸 + 𝑞 1 + 𝑞 − 0 , 𝑞  = 𝑞 + 𝑞 2 − 𝑞 𝑛  𝑙 = 0 ⎛ ⎜ ⎜ ⎝ 𝑛 𝑙 ⎞ ⎟ ⎟ ⎠  𝐸 𝑙 , 𝑞 = 𝑞 + 𝑞 2   𝐸 − 𝑞 𝑞  + 1 𝑛 = 𝑞 + 𝑞 2 +  𝐸 𝑛 , 𝑞 [ 2 ] − 𝑞 𝑞 𝛿 0 , 𝑛 . ( 2 . 2 3 ) Therefore, by ( 2.23 ), we obtain the following theorem. Theorem 2.5. For 𝑛 ∈ ℕ , one has 𝑞 2  𝐸 𝑛 , 𝑞 ( 2 ) = 𝑞 + 𝑞 2 +  𝐸 𝑛 , 𝑞 . ( 2 . 2 4 ) For 𝑛 ∈ ℤ + , we have  𝐸 𝑛 , 𝑞 − 1 (  1 − 𝑥 ) = ℤ 𝑝  1 − 𝑥 + 𝑥 1  𝑛 𝑑 𝜇 − 𝑞 − 1  𝑥 1  = ( − 1 ) 𝑛  ℤ 𝑝  𝑥 1  + 𝑥 𝑛 𝑑 𝜇 − 𝑞  𝑥 1  = ( − 1 ) 𝑛  𝐸 𝑛 , 𝑞 ( 𝑥 ) . ( 2 . 2 5 ) Therefore, by ( 2.25 ), we obtain the following theorem. Theorem 2.6. For 𝑛 ∈ ℤ + , one has  𝐸 𝑛 , 𝑞 − 1 ( 1 − 𝑥 ) = ( − 1 ) 𝑛  𝐸 𝑛 , 𝑞 ( 𝑥 ) . ( 2 . 2 6 ) From ( 2.20 ), we have  ℤ 𝑝 ( 1 − 𝑥 ) 𝑛 𝑑 𝜇 − 𝑞 ( 𝑥 ) = ( − 1 ) 𝑛  ℤ 𝑝 ( 𝑥 − 1 ) 𝑛 𝑑 𝜇 − 𝑞 ( 𝑥 ) = ( − 1 ) 𝑛  𝐸 𝑛 , 𝑞 ( − 1 ) . ( 2 . 2 7 ) By Theorem 2.6 and ( 2.27 ), we get  ℤ 𝑝 ( 1 − 𝑥 ) 𝑛 𝑑 𝜇 − 𝑞 (  𝐸 𝑥 ) = 𝑛 , 𝑞 − 1 ( 2 ) = 1 + 𝑞 + 𝑞 2  𝐸 𝑛 , 𝑞 − 1 i f 𝑛 > 0 . ( 2 . 2 8 ) Therefore, by ( 2.28 ), we obtain the following theorem. Theorem 2.7. For 𝑛 ∈ ℕ , one has  ℤ 𝑝 ( 1 − 𝑥 ) 𝑛 𝑑 𝜇 − 𝑞 ( 𝑥 ) = 1 + 𝑞 + 𝑞 2  𝐸 𝑛 , 𝑞 − 1 . ( 2 . 2 9 ) Let 𝐶 ( ℤ 𝑝 ) be the space of continuous functions on ℤ 𝑝 . For 𝑓 ∈ 𝐶 ( ℤ 𝑝 ) , 𝑝 -adic analogue of Bernstein operator of order 𝑛 for 𝑓 is given by 𝔹 𝑛 ( 𝑓 ∣ 𝑥 ) = 𝑛  𝑘 = 0 𝐵 𝑘 , 𝑛  𝑘 ( 𝑥 ) 𝑓 𝑛  = 𝑛  𝑘 = 0 𝑓  𝑘 𝑛  ⎛ ⎜ ⎜ ⎝ 𝑛 𝑘 ⎞ ⎟ ⎟ ⎠ 𝑥 𝑘 ( 1 − 𝑥 ) 𝑛 − 𝑘 , ( 2 . 3 0 ) where 𝑛 , 𝑘 ∈ ℤ + (see [ 1 , 6 , 7 ]). For 𝑛 , 𝑘 ∈ ℤ + , 𝑝 -adic Bernstein polynomial of degree 𝑛 is defined by 𝐵 𝑘 , 𝑛 ⎛ ⎜ ⎜ ⎝ 𝑛 𝑘 ⎞ ⎟ ⎟ ⎠ 𝑥 ( 𝑥 ) = 𝑘 ( 1 − 𝑥 ) 𝑛 − 𝑘 , 𝑥 ∈ ℤ 𝑝 ( 2 . 3 1 ) (see [ 1 , 6 , 7 ]). Let us take the fermionic 𝑝 -adic 𝑞 -integral on ℤ 𝑝 for one Bernstein polynomials in ( 2.31 ) as follows:  ℤ 𝑝 𝐵 𝑘 , 𝑛 ( 𝑥 ) 𝑑 𝜇 − 𝑞 ⎛ ⎜ ⎜ ⎝ 𝑛 𝑘 ⎞ ⎟ ⎟ ⎠  ( 𝑥 ) = ℤ 𝑝 𝑥 𝑘 ( 1 − 𝑥 ) 𝑛 − 𝑘 𝑑 𝜇 − 𝑞 = ⎛ ⎜ ⎜ ⎝ 𝑛 𝑘 ⎞ ⎟ ⎟ ⎠ ( 𝑥 ) 𝑛 − 𝑘  𝑙 = 0 ⎛ ⎜ ⎜ ⎝ 𝑙 ⎞ ⎟ ⎟ ⎠ ( 𝑛 − 𝑘 − 1 ) 𝑙  ℤ 𝑝 𝑥 𝑘 + 𝑙 𝑑 𝜇 − 𝑞 ( = ⎛ ⎜ ⎜ ⎝ 𝑛 𝑘 ⎞ ⎟ ⎟ ⎠ 𝑥 ) 𝑛 − 𝑘  𝑙 = 0 ⎛ ⎜ ⎜ ⎝ 𝑙 ⎞ ⎟ ⎟ ⎠ 𝑛 − 𝑘 ( − 1 ) 𝑙  𝐸 𝑘 + 𝑙 , 𝑞 . ( 2 . 3 2 ) By simple calculation, we easily get  ℤ 𝑝 𝐵 𝑘 , 𝑛 ( 𝑥 ) 𝑑 𝜇 − 𝑞 (  𝑥 ) = ℤ 𝑝 𝐵 𝑛 − 𝑘 , 𝑛 ( 1 − 𝑥 ) 𝑑 𝜇 − 𝑞 ( = ⎛ ⎜ ⎜ ⎝ 𝑛 𝑘 ⎞ ⎟ ⎟ ⎠ 𝑥 ) 𝑘  𝑙 = 0 ⎛ ⎜ ⎜ ⎝ 𝑘 𝑙 ⎞ ⎟ ⎟ ⎠ ( − 1 ) 𝑘 + 𝑙  ℤ 𝑝 ( 1 − 𝑥 ) 𝑛 − 𝑙 𝑑 𝜇 − 𝑞 = ⎛ ⎜ ⎜ ⎝ 𝑛 𝑘 ⎞ ⎟ ⎟ ⎠ ( 𝑥 ) 𝑘  𝑙 = 0 ⎛ ⎜ ⎜ ⎝ 𝑘 𝑙 ⎞ ⎟ ⎟ ⎠ ( − 1 ) 𝑘 + 𝑙  1 + 𝑞 + 𝑞 2  𝐸 𝑛 − 𝑙 , 𝑞 − 1  = ⎛ ⎜ ⎜ ⎝ 𝑛 𝑘 ⎞ ⎟ ⎟ ⎠ 𝑘  𝑙 = 0 ⎛ ⎜ ⎜ ⎝ 𝑘 𝑙 ⎞ ⎟ ⎟ ⎠ ( − 1 ) 𝑘 + 𝑙 𝑞 2  𝐸 𝑛 − 𝑙 , 𝑞 − 1 + [ 2 ] 𝑞 ⎛ ⎜ ⎜ ⎝ 𝑛 𝑘 ⎞ ⎟ ⎟ ⎠ ( − 1 ) 𝑘 𝛿 0 , 𝑘 i f 𝑛 > 𝑘 . ( 2 . 3 3 ) Therefore, by ( 2.32 ) and ( 2.33 ), we obtain the following theorem. Theorem 2.8. For 𝑛 ∈ ℤ + with 𝑛 > 𝑘 > 0 , one has 𝑛 − 𝑘  𝑙 = 0 ⎛ ⎜ ⎜ ⎝ 𝑙 ⎞ ⎟ ⎟ ⎠ 𝑛 − 𝑘 ( − 1 ) 𝑙  𝐸 𝑘 + 𝑙 , 𝑞 = 𝑘  𝑙 = 0 ⎛ ⎜ ⎜ ⎝ 𝑘 𝑙 ⎞ ⎟ ⎟ ⎠ ( − 1 ) 𝑘 + 𝑙 𝑞 2  𝐸 𝑛 − 𝑙 , 𝑞 − 1 . ( 2 . 3 4 ) In particular, 𝑘 = 0 , we get 𝑛  𝑙 = 0 ⎛ ⎜ ⎜ ⎝ 𝑛 𝑙 ⎞ ⎟ ⎟ ⎠ ( − 1 ) 𝑙  𝐸 𝑙 , 𝑞 = 𝑞 2  𝐸 𝑛 , 𝑞 − 1 + [ 2 ] 𝑞 . 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