Explicit relations for the modified degenerate Apostol-type polynomials

Transcription

1 Araştırma Makalesi BAUN Fen Bil. Enst. Dergisi, 20(2), , (2018) DOI: /baunfbed J. BAUN Inst. Sci. Technol., 20(2), , (2018) Explicit relations for the modified degenerate Apostol-type polynomials Burak KURT * Akdeniz University Faculty of Education, Department of Mathematics, Antalya. Geliş Tarihi (Recived Date): Kabul Tarihi (Accepted Date): Abstract Recently, the degenerate Bernoulli numbers polynomials the degenerate Euler numbers polynomials have been studied by several authors. In this paper, we consider the modified Apostol-Bernoulli polynomials the modified Apostol-Euler polynomials. We give explicit relation for the modified degenerate Bernoulli polynomials the modified degenerate Euler polynomials. Also, we prove some identities between the modified Apostol-Bernoulli polynomials the degenerate Stirling numbers of two kinds. Keywords: Bernoulli polynomials numbers, Euler polynomials numbers, Modified Bernoulli numbers polynomials, Modified Euler numbers polynomials, Degenerate Stirling numbers of the second kind, Degenerate µ-multiple sums, Degenerate µ-multiple alternating sums. Modifiye dejenere Apostol-tipi polinomlar için kesin bağıntılar Özet Son yıllar da dejenere Bernoulli sayıları, polinomları ve dejenere Euler sayıları, polinomlarını birçok yazarlar tarafından çalışılıyor. Bu makale de modifiye Apostol- Bernoulli polinomları ve modifiye Apostol-Euler polinomlarını tanımladık. Modifiye dejenere Bernoulli polinomları ve modifiye dejenere Euler polinomları için kesin bağıntı verdik. Ayrıca, ikinci çeşit dejenere Stirling sayıları ve modifiye Apostol- Bernoulli polinomları arasında bazı özellikler ispatlı. Anahtar kelimeler: Bernoulli polinomları ve sayıları, Euler polinomları ve sayıları, Modifiye Bernoulli polinomları ve sayıları, Modifiye Euler polinomları ve sayıları, İkinci Çeşit Dejenere Stirling sayıları, Dejenere µ-katlı toplamlar. * Burak KURT, burakkurt@akdeniz.edu.tr, 401

2 KURT B. 1. Introduction As usual, throughout this paper, N denotes the set of natural numbers, N 0 denotes the set of nonnegative integers, R denotes the set of real numbers C denotes the set of complex numbers. We begin by introducing the following definitions notations (see also [14-17]). It is well known the Bernoulli polynomials Euler polynomials are defined by the generating functions respectively =0, < 2 (1)! = 1 =0, <. (2) = 2! +1 When x=0, = 0 = 0 are called the Bernoulli numbers the Euler numbers respectively. The generalized Apostol-Bernoulli polynomials () ; of order 0 the generalized Apostol-Euler polynomials () ; of order 0 are defined by the following generating functions (see, for detail [9, 14-17]) () ; =0 =! 1, < 2 h = 1, < h 1 (3) () ; =! 2 =0 +1, < h = 1, < log ( ) h 1. (4) Carlitz in [1, 2] defined degenerate Bernoulli polynomials which are given by the generating function to be / 1 + / =, (5)! when x=0, () = 0 are called the degenerate Bernoulli numbers. From (5), we can easily derive the following equations =, > 0 (i) where = 1, = = ( + 1). (ii) Dolgy et al. [3] studied the following modified degenerate Bernoulli polynomials, () which are different from Carlitz s degenerate Bernoulli polynomials generated by (5) as follows 402

3 BAUN Fen Bil. Enst. Dergisi, 20(2), , (2018) / 1 + / = (6)!, () which, in the special case when = 0,, 0,. We have the modified degenerate Bernoulli numbers,. It is easily observed from the generating function (6) that lim,() = lim! Thus, by applying (7), we find that lim, = (). 1 + / / = = (). (7)! Kwon et al. in [8] studied the analogously modified degenerate Euler polynomials generated by / 1 + / = (8)!, () which in the special case when = 0,, 0, reduces to the generating function for the modified degenerate Euler numbers,. Motivated essentially by each of these works [3] [8], we consider investigate the generalized higher order modified degenerate Apostol-Bernoulli polynomials () the generalized higher order modified degenerate Apostol-Euler, polynomials by means of following generating functions (),! (),! = / () 1 + / (9) = / () 1 + / (10) respectively. Here in what follows where in particular, for x=0 in (9) (10), we have the generalized higher order modified degenerate Apostol- Bernoulli numbers,() the generalized higher order modified degenerate Apostol-Euler numbers,(), respectively. By applying to the generating functions (9) (10), we get lim (), =,() = (, ) 403

4 KURT B. lim, () =, () = (, ) respectively. A degenerate version of the Stirling number, of the second kind is defined by generating function! 1 + / 1 = In terms of the multinomial coefficients given by,,,!!!!,,. (11)! the -multiple power sums were defined by Luo [12] as follows ; = which readily yields,,, (12) = () ; (13)! where. Similarly, the -multiple alternating power sums were defined by Luo [13] as follows ; = 1 which readily yields.,,, (14) = () ; (15)! where. From (13) (15), we define the -multiple degenerate power sums the -multiple degenerate alternating power sums by means of the following equations / / = (), ; (16)! 404

5 BAUN Fen Bil. Enst. Dergisi, 20(2), , (2018) / / = (), ;. (17) Keeping in view many of the above-mentioned other related investigation by Carlitz (see [1, 2]), Dolgy et al. [3], Kim et al. [5], He et al. [6], Kim [7], Kwon et al. [8], Kurt [9, 10], Liu Wang [11], Luo [12, 13], Srivastava [16], Kurt [18], Yang [21], Young [22]. We systematically study the above defined the generalized higher order modified degenerate Apostol-Bernoulli polynomials the generalized higher order modified degenerate Apostol-Euler polynomials. In particular, we give some explicit relation between the modified degenerate Bernoulli polynomials the modified degenerate Euler polynomials. Also, we prove identities for the modified degenerate Apostol-Bernoulli polynomials modified degenerate Apostol-Euler polynomials.! 2. Explicit relations for the modified degenerate Bernoulli Euler polynomials In this section, we give some explicit relationships for the modified degenerate Bernoulli the modified degenerate Euler polynomials. We prove some identitites for these polynomials. Also, by using the equation (9) (10), we can obtain the following relations:, + 1, =,, + 1 +, =, (i) (ii) where r=1 in (9) (10) + =,,, (iii) = (),, (iv),,, = 2,,, (v) where r=1 in (9) (10). Theorem 1. There is the following relation between the modified degenerate Bernoulli polynomials the degenerate Stirling numbers of the second kind: =!!,,. (18)! Proof. From (5) (11),! = / 1 + / 405

6 KURT B. = / 1 + / / =! 1 + / /!! /,, =!,, =!,,!! =!!!!!!,,!. Since the right h of this equality to n=k is zero, comparing both sides of this equality, we have (18). Theorem 2. The following relation holds true! =,,. (19) Proof. By using the identities / = 1 + / () = =! / 1 + / =! /!!! =,, = 1 + /! /!,, Comparing the coefficients of both sides of equation, we have result. Theorem 3. There is the following relation between the degenerate Bernoulli number the degenerate Stirling numbers of the second kind as: () =! Proof. From (5), for x=0 0 =! / = 1 + / / = = =! /!!.! ( 1), +,. (20)! 1 =! 1,,! =! 1,,!! =!! ( 1), +, ()! 1 + / 1!!!!. 406

7 BAUN Fen Bil. Enst. Dergisi, 20(2), , (2018) Comparing the coefficients of both sides of, we have (20). Theorem 4. The degenerate Euler polynomials satisfy the following relation! = 2.. (21) Proof. By using the following identitites (8), = / / / / we write / / / From last equality, we write as = / / / /. / 1 + / = / / 1 + / 21 + / =! 1! 2!.! By using Cauchy product comparing the coefficient, we have result. 3. Some symmetry identities for the modified degenerate Apostol-Bernoulli polynomials In this section, by using µ-multiple power sums, we give some symmetry identities for the modified degenerate Apostol-Bernoulli polynomials. Theorem 5. There is the following relation between the modified degenerate Apostol- Bernoulli polynomials the modified µ-multiple power sums: 1,,, + + = 1,,, + + Proof. Let = / / / / / (22) 407

8 KURT B. = / 1 + /. By using (9) (13) for l=1, = 1,, 1 1, + + Using Cauchy product, we have =!. 1,, 1, + +. (23)! In similar manner, = / / / / /. From (9) (13), we write! 1,, 1, + + =!. (24) By comparing the coefficients of in (23) (24), we prove the theorem.! Theorem 6. For all,,, we have the following symmetry identities = Proof. Let (), (), (),,, (),, (25) h = / / / / / = =, 1 + 1! (),, 1 =!,, 1 (),, In a similar manner,, / 1 +,!. (26)! 408

9 BAUN Fen Bil. Enst. Dergisi, 20(2), , (2018) h = / / / / / = =, 1 +! 1 (),, =!,, 1 (),,! / 1 +,!. (27) By comparing the coefficients of in the above equation (26) (27), we get (25).! 4. Some symmetry identities for the modified degenerate Apostol-Euler polynomials In this section, by using -multiple power sums, we give some symmetry identities for the modified degenerate Apostol-Euler polynomials. Theorem 7. Let a b be positive integers with the same parity. Then, 1, (; ) =, 1, (; ). (28) Proof. Let h = / / / /. From (10) (17) for l=1, we have h =,! 1, (; )! =, 1, (; ). (29) Since 1 = 1, the expression for! h = / / / /. is symmetric in a b. Then we obtain the following power series for h by symmetry h =,! 1, (; )! =, 1, (; ). (30)! 409

10 KURT B. Equating the coefficients of in (29) (30) for h gives us the desired result.! Theorem 8. Let a b be positive integers with the same parity. Then (), = (), Proof. Let = / / / / (), (; ), (), (; ), (31) 1 + / = / 1 + / / / / =, +1 1 = 1! +1 1, 1 1, (; ) / 1 +! =0,, (; ),. (32)! Since 1 = 1, the expression for is symmetric in a b. In a similar manner, we have = / 1 + / / / = +1, = 1 1! +1 1, / 1 1, (; ) / 1 +! =0,, (; ),. (33)! Equating the coefficients of in (32) (33) for gives us the desired result.! Theorem 9. Let p, l, a, b n be positive integers a b be of the same parity. Then, 1, (; ) = 2, 2 + 1, 2. (34) Proof. = / / / /.!! 410

11 BAUN Fen Bil. Enst. Dergisi, 20(2), , (2018) From (9) (17), we have =,! 1, (; )! =, =0 1, (; ).! On the other h, we write the function as / / + / / / / = =, 2 = 2!, 2, 2 + 1, 2 + Equating the coefficients of, we obtain (34).!!.! Acknowledgement The present investigation was supported, by the Scientific Research Project Administration of Akdeniz University. References [1] Carlitz L., A note on Bernoulli Euler polynomials of the second kind, Scripta Mathematica, 25, , (1961). [2] Carlitz L., Degenerate Stirling, Bernoulli Eulerian numbers, Utilitas Mathematica, 15, 51-88, (1979). [3] Dolgy D.V., Kim T., Known H.-In Seo J.J., On the modified degenerate Bernoulli polynomials, Advanced Studies in Contemporary Mathematics, 26, 1-9, (2016). [4] He Y., Araci S. Srivastava H.M., Some new formulas for the products of the Apostol type polynomials, Advances in Difference Equations, 2016, Article ID 287, 1-18, (2016). [5] Kim T., Kim D.S. Kwon H.-In, Some identities relating to degenerate Bernoulli polynomials, Filomat, 30, , (2016). [6] Kim T. Seo J.J., On generalized degenerate Bernoulli numbers polynomials, Applied Mathematical Sciences, 9, 120, , (2015). [7] Kim T., Degenerate Bernoulli polynomials associated with p-adic invariant integral on Z_{p}, Advanced Studies in Contemporary Mathematics, 25, 3, , (2015). [8] Kwon H.-In, Kim T. Seo J.J., Modified degenerate Euler polynomials, Advanced Studies in Contemporary Mathematics, 26, , (2016). [9] Kurt B., Some relationships between the generalized Apostol-Bernoulli Apostol-Euler polynomials, Turkish Journal of Analysis Number Theory, 1, 1-7, (2013). 411

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