On the New q-extension of Frobenius-Euler Numbers and Polynomials Arising from Umbral Calculus

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1 Adv. Studies Theor. Phys., Vo. 7, 203, no. 20, HIKARI Ltd, On the New -Extension of Frobenius-Euer Numbers and Poynomias Arising from Umbra Cacuus Dae San Kim Department of Mathematics, Sogang University Seou 2-74, Repubic of Korea Taeyun Kim Department of Mathematics, Kwangwoon University Seou 39-70, Repubic of Korea Jongjin Seo Department of Appied Mathematics Puyong Nationa University Pusan , Repubic of Korea Copyright c 203 Dae San Kim, Taeyun Kim and Jongjin Seo. This is an open access artice distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the origina wor is propery cited. Abstract. In this paper, we consider the new -extension of Frobenius- Euer numbers and poynomias and we derive some interesting identities from the orthogonaity type properties for the new -extension of Frobenius-Euer poynomias. Finay we suggest one open uestion reated to orthogonaity of poynomias. Mathematics Subject Cassification: B68, S80 Keywords: umbra agebra, umbra cacuus, n th ordinary Frobenius- Euer poynomias

2 978 Dae San Kim, Taeyun Kim and Jongjin Seo. Introduction For [0, ], we define the shifted factorias by (a : 0,(a : n n i0 ( ai, (a : i0 ( ai. If x is cassica object, such as a compex number, its version is defined by [x] x. As is we nown, the exponentia function is given by z n e (z [n]!, (see [5], [7], (. (( z : where z C with z <. The derivative is defined by D f(x f(x f(x, (see [5], [8], [9], [0]. (.2 ( x df (x dx. Note that im D f(x The definite integra is given by x f(td t ( f( a xx a, (see [4], [7], [0]. (.3 0 From (.2 to (.3, we have x a0 t n d t x n+, D x n [n] x n. (.4 0 [n +] The Bernoui poynomias of Hegazi and Mansour are defined by the generating function to be B n, (x tn [n]! t e (t e (xt, (see [4]. (.5 In the specia case, x 0,B n, (0 B n, are caed the n th Bernoui numbers. By(.5, we get no ( B n, (x tn [n]! B, t []! 0 ( n ( n 0 ( m0 B, x n where ( n [n]!, [n] []![n ]!![n] [n ] [2] []. x m [m]! tm t n [n]! (.6

3 On the new -extension of Frobenius-Euer numbers and poynomias 979 Thus, from (.6, we have n ( n B n, (x B, x n. (.7 0 In [.6], the new extension of Euer poynomias are aso defined by the generating function to be 2 e (t+ e (xt E n, (x tn [n]!. (.8 In the specia case, x 0, E n, (0 E n, are caed the n th Euer numbers. By (.8, we easiy get n ( n E n, (x E, x n, 0 (see [7]. (.9 More than five decades age, Caritz(see [8],[9] defined a extension of Bernoui and Euer poynomias. In a recent paper(see [0], B.A. Kupershmidt constructed refection symmetries of Bernoui numbers and poynomias. From the methods of B.A. Kupershmidt, Hegazi and Mansour derived some interesting identities and properties reated to Bernoui and Euer poynomias. Recenty, severa authors have studied various extention of Bernoui, Euer and Genocchi poynomias(see [],[2],[4]-[4]. Let F be the set of a forma power series in variabe t over C with { F f(t } a t a C. (.0 [] 0 Let P C[x] and et P be the vector space of a inear functionas on P. < L p(x > denotes the action of inear functiona L on the poynomias p(x, and it is we now that the vector space operation on P defined by <L+M p(x >< L p(x > + <M p(x >, < cl p(x > c<l p(x >, where c is compex constant. For f(t 0 a [] t! F, Let us define the inear functiona on P by setting By (., we get <f(t x n > a n (n 0. (. <t x n >[n]! δ n,, (n, 0, (.2 where δ n, is Kronecer s symbo. Let f L (t 0 <L x > t []!. Then, by (.2, we get <f L (t x n >< L x n >. (.3

4 980 Dae San Kim, Taeyun Kim and Jongjin Seo Additionay, the map L f L (t is a vector space isomorphism from P onto F. Henceforth, F denotes both the agebra of forma power series in t and the vector space of a inear functionas on P, and so an eement f(t of F wi be thought as a forma power series and a inear functiona. We ca F the umbra agebra. The umbra cacuus is the study of umbra agebra. By (. and (., we easiy see that <e (yt x n > y n and so <e (yt p(x > p(y for p(x P. The order O(f(t of the power series f(t 0 is the smaest positive integer for which a does not vanish. If O(f(t 0, then f(t is caed an invertibe series. If O(f(t, then f(t is caed a deta series(see[3], [4]. For f(t F(t, p(x P, we have f(t <f(t t > 0 t []!, p(x 0 <t p(x > x []!. (.4 Thus, by (.4, we get p ( (0 < t p(x >, < p ( (x > p ( (0, (.5 where p ( (x D p(x (see[3], [4]. From (.5, we have t p(x p ( (x D p(x. (.6 For f(t,g(t F with O(f(t, O(g(t 0, there exists a uniue seuence S n (x(degs n (x n of poynomia such that <g(tf(t S n (x > [n]! δ n,, (n, 0. The seuence S n (x is caed the Sheffer seuence for (g(t,f(t which is denoted by S n (x (g(t,f(t. For S n (x (g(t,f(t,we have g(f(t e (yf(t 0 S (y []! t, for a y C, (.7 where f(t is the compositiona inverse of f(t (see[3]. Many researchers tod that the properties of the extension of Frobenius- Euer poynomias are vauabe and worthwhie in the areas of both number theory and mathematica physics (see[],[2],[4]-[2]. In this paper, we consider new approach to extension of Frobenius-Euer numbers and poynomias which are derived from umbra Cacuus and we give some interesting identities of our Frobenius-Euer numbers and poynomias. Finay, we suggest one open uestion reated to orthogonaity of Caritz s Bernoui poynomias.

5 On the new -extension of Frobenius-Euer numbers and poynomias extension of Frobenius-Euer poynomias In this section, we assume that λ C with λ. Now, we consider new extension of Frobenius-Euer poynomias which are derived form the generating function as foows: λ e (t λ e (xt H n, (x λ tn [n]!. (2. In the specia case, x 0, H n, (0 λ H n, (λ are caed the n th Frobenius-Euer numbers. Note that im H n, (x λ H n (x λ, where H n (x λ are the n th ordinary Frobenius-Euer poynomias. By (2., we easiy see that H n, (x λ tn [n]! Thus, from (2.2, we have ( ( H, (λ t []! 0 m0 ( n ( n 0 H, (λx n x m [m]! tm t n [n]!. (2.2 n ( n H n. (x λ H, (λx n. (2.3 0 By the definition of Frobenius-Euer numbers, we get ( λ H, (λ t (e (t λ []! 0 { n ( } n t n H n m, (λ m mo [n]! λ H n, (λ tn [n]! {H n, ( λ λh n, (λ} tn [n]!. By comparing the coefficients on the both sides, we get { λ, if n 0 H 0, (λ,h n, ( λ λh n, (λ 0, if n>0. Therefore, by (2.3 and (2.5, we obtain the foowing theorem. Theorem 2.. For n 0, we have n H n, (x λ 0 ( n H n, (λx. (2.4 (2.5

6 982 Dae San Kim, Taeyun Kim and Jongjin Seo Moreover, H 0, (λ, H n, ( λ λh n, (λ ( λδ 0,n. For exampes, H 0, (λ,h, (λ,h λ 2,(λ From (.7 and (2., we have and Thus, by (2.7, we get H n, (x λ λ e (t λ e (xt λ+ ( λ 2,. ( e (t λ λ,t, (2.6 H n, (x λ tn [n]!. (2.7 λ e (t λ xn H n, (x λ, (n 0. (2.8 From (.4, (.6 and (2.8, we have th n, (x λ λ λ e (t λ txn [n] e (t λ xn (2.9 [n] H n, (x λ. Therefore, by (2.8 and (2.9, we obtain the foowing emma. Lemma 2.2. For n 0, we have H n, (x λ λ e (t λ xn, th n, (x λ [n] H n, (x λ. By (2.9, we get e (t λ λ t H n, (x λ []! λ []! λ ( n ( n e (t λ H n, (x λ {H n, ( λ λh n, (λ}. (2.0 From (2.6, we have e (t λ λ t H n, (x λ [n]! δ n,. (2. By (2.0 and (2., we get

7 On the new -extension of Frobenius-Euer numbers and poynomias λ (H n,( λ λh n, (λ n ( n H, (λ+( λh n, (λ, 0 where n, Z 0 with n>. Thus, from (2.2, we have (2.2 H n, (λ λ n 0 ( n where n, Z 0 with n>. Therefore, by (2.3, we obtain the foowing theorem. Theorem 2.3. For n, we have x+y H n, (λ λ n 0 ( n H, (λ. H, (λ, (2.3 From (.4 and (2.3, we have n ( n + H n, (u λd u H n, (λ { (x + y + x +} x [n +] + 0 n+ ( n + H n+, (λ { (x + y x } [n +] 0 {H n+, (x + y λ H n+, (x λ}. [n +] Thus, by (2.4, we get e (t e (t H n, (x λ th n+, (x λ t [n +] t e (t H n+, (x λ [n +] {H n+, ( λ H n+, (λ} [n +] 0 H n, (u λd u. Therefore, by (2.5, we obtain the foowing theorem. (2.4 (2.5

8 984 Dae San Kim, Taeyun Kim and Jongjin Seo Theorem 2.4. For n 0, we have e (t H n, (x λ t Let 0 H n, (u λd u. P n {p(x C[x] deg p(x n}, (2.6 be the (n + dimensiona vector space over C. Forp(x P n, et us assume that p(x n C H, (x λ, (n 0. (2.7 0 Then, by (2.6, (2. and (2.7, we get ( e (t λ t p(x λ n ( e (t λ C t H, (x λ λ n C []! δ, []! C. By (2.8, we get C ( e (t λ t p(x []! λ { p ( ( λp ( (0 }, ( λ[]! 0 0 (2.8 (2.9 where p ( (x D p(x. Therefore, by (2.7 and (2.9, we obtain the foowing theorem. Theorem 2.5. For p(x P n, et p(x n 0 C H, (x λ Then we have { C p ( ( λp ( (0 }, []!( λ where p ( (x D p(x. Let us tae p(x B n, (x with B n, (x p(x n C H, (x λ. (2.20 0

9 On the new -extension of Frobenius-Euer numbers and poynomias 985 Then, by Theorem 2.5, we get C []!( λ {[n] [n ] [n +] B n, ( λ[n] [n +] B n, } ( n λ {B n,( λb n, }. (2.2 Therefore, by (2.2, we obtain the foowing theorem. Theorem 2.6. For n, 0, with n >0, we have B n, (x n ( n {B n, ( λb n, } H, (x λ. λ 0 Let us tae p(x x n with From (.5, we have Thus, we have x n λ x n p(x C λ n λ n ( n 0 ( n 0 n C H, (x λ. ( ( n λ λ H, (x λ By the same method, we easiy see that E n, (x λ n 0 ( n 0 n. (2.23 λ λ H n,(x λ H, (x λ H n, (x λ. ( n {E n, ( λe n, } H, (x λ. The extension of Frobenius-Euer poynomias of order r (r N are defined by the generating function to be ( r λ e (xt e (t λ H n, (r tn (x λ [n]!. (2.24 In the specia case, x 0, H n,(0 (r λ H n,(λ (r are caed the n th Frobenius-Euer numbers of order r.

10 986 Dae San Kim, Taeyun Kim and Jongjin Seo From (2.24, we can derive the foowing euation (2.25: n ( n H n, (r (x λ H (r n, (λx. (2.25 by (.7 and (2.24, we get 0 (( r H n, (r (x λ e (t λ,t. (2.26 λ Thus, from (2.26, we have ( r e (t λ t H (r λ n,(x λ [n]! δ n,. (2.27 For p(x P n, et us assume that p(x n 0 CH r (r, (x λ, (n 0. (2.28 From (2.27 and (2.28, we can derive the foowing euation (2.29: ( r e (t λ t p(x λ Thus, by (2.29, we get C r []! n C r 0 ( r e (t λ t H (r, λ (x λ n C r []!δ, C[] r!. 0 ( r e (t λ t p(x []!( λ r []!( λ r []!( λ r λ r ( r j j0 r ( r j j0 0 r ( λ r j (e (t j t p(x ( λ r j j0 + + j j ( r j ( λ r j p (+ (x [ ]! [ j ]! [ ]! [ j ]! p(+ (0. (2.29 (2.30 Therefore, by (2.28 and (2.30, we obtain the foowing theorem.

11 On the new -extension of Frobenius-Euer numbers and poynomias 987 Theorem 2.7. For p(x P n, et p(x n Then we have C r []!( λ r 0 where p (+ (x D + p(x. r From (.7 and (2.30, we have j0 + + j 0 Cr H(r, (x λ. ( r j ( λ r j [ ]! [ j ]! p(+ (0, ( r H n, (r tn λ (x λ [n]! e (xt. (2.3 e (t λ Thus, from (2.3, we have. and th (r ( r λ n, (x λ x n, (n 0, (2.32 e (t λ H (r ( r ( r λ λ n, (x λ tx n [n] x n e (t λ e (t λ [n] H (r n,(x λ. Indeed, ( r λ e (yt x n e (t λ and ( r λ x n e (t λ where ( n i,,i r [n]! [i ]! [i r]!. n H n, (r (y λ ( m0 i + +i rn i + +i rn i + +i rm From (2.34, we note that ( r λ x n e (t λ 0 (2.33 ( n H (r n, (λy, (2.34 H i,(λ H ir,(λ t m x n [i ]! [i r ]! [n]! [i ]! [i r ]! H i,(λ H ir,(λ ( n H i,(λ H ir,(λ, i,,i r (2.35 H n, (r (λ, (n 0. (2.36

12 988 Dae San Kim, Taeyun Kim and Jongjin Seo Therefore, by (2.35 and (2.36, we obtain the foowing theorem. Theorem 2.8. For n 0, we have H n, (r (λ ( n H i,(λ H ir,(λ. i,,i r i + +i rn Let us tae p(x H (r n,(x λ P n with H (r n,(x λ p(x n 0 C H, (x λ. Then, by Theorem2.5, we get C (e (t λ t p(x ( λ[]! ( n e (t λ H (r n, λ (x λ ( n { } H (r (r n, ( λ λh n, λ (λ. From (2.24, we can readiy derive the foowing euation (2.38: { H (r n, ( λ λh n,(λ } t n [n]! ( r ( λ λ (e (t λ ( λ e (t λ e (t λ ( λ H n, (r (λ tn [n]!. r By comparing the coefficients on the both sides of (2.38, we get (2.37 (2.38 H (r n, (r (r ( λ λh n, (λ ( λh n, (λ. (2.39 Therefore, by (2.37 and (2.39, we obtain the foowing theorem. Theorem 2.9. For n Z 0 and r N, we have n ( n H n,(x (r λ H (r n, (λh,(x λ. 0 Let us assume that p(x H n, (x λ P n with H n, (x λ p(x Then, by Theorem 2.7, we get n 0 C r H(r, (x λ. (2.40

13 On the new -extension of Frobenius-Euer numbers and poynomias 989 C r n ( λ r []! m0 n ( λ m0 ( λ r r ( r ( m 0 i + +i m r 0 ( λ r i,,i ( r ( λ r [n] [n m +] H n m, (λ [m]! i + +i m ( m i,,i [m + ]! [m] []! [n] [n m +] H n m, (λ [m + ]! n r ( r m0 0 i + +i m ( m + ( m i,,i m ( n ( λ r H n m, (λ. m + Therefore, by (2.40 and (2.4, we obtain the foowing theorem. Theorem 2.0. For n, r 0, we have H n, (x λ { n r ( λ r m0 0 i + +i m H (r, (x λ. ( r n 0 ( λ r ( ( m m + i,,i m (2.4 ( } n H n m, (λ m + Remar. By the same method, we can see that B (r n,(x ( λ r n 0 H (r, (x λ. { n r m0 0 i + +i m ( r ( ( m m + ( λ r i,,i m ( } n B (r n m, m +

14 990 Dae San Kim, Taeyun Kim and Jongjin Seo As is we nown, the Cartz s Bernoui poynomias are defined by the generating function to be t n f(t, x β n, n! (2.42 t 2m e [m+x]t +( m e [m+x]t. m0 Thus, we note that f(t, x is an invertibe series. From (2.42, we note that β n, (x ( n n 0 m0 ( n ( x +, (see [3]. (2.43 [ +] In the specia case, x 0,β n, β n, (0 are caed the n th Caritz s Bernoui numbers. By (2.42 and (2.43, we readiy see that n ( n β n, (x x β, [x] n (f(t, x,t. 0 Open Probem. Let P n, {p([x] C[[x] ] deg p([x] n} For p([x] P n,, et us assume that p([x] n 0 C,β, (x. Determine the coefficient C, References. A. Bayad, T. Kim Higher recurrences for Aposto-Bernoui-Euer numbers, Russ. J. Math. Phys. 9(202, no., L. Caritz, Bernoui numbers and poynnmias, Due Math. J. 5(948, R. Dere, Y. Simse, Appications of umbra agebra to some specia poynomias, Adv. Stud. Contemp. Math. 22(202, A. S. Hegazi, M. Mansour, A note on -Bernoui numbers and poynomias, J. Noninear Math. Phys. 3 (2006, no., A.S. Hegazi, M. Mansour, Weight function approach to -difference euations for the -hypergeometric poynomias, Internat. J. Theoret. Phys. 43 (2004, no., A. Khrenniov, Quantum probabiities from a mathematica mode of threshod detection of cassica random waves, J. Phys. A 45 (202, no. 2, 2530, 24 pp. 7. T. Kim, -generaized Euer numbers and poynomias, Russ. J. Math. Phys. 3 (2006, no. 3, T. Kim, Power series and asymptotic series associated with the -anaog of the twovariabe p-adic L-function, Russ. J. Math. Phys. 2 (2005, no. 2,

15 On the new -extension of Frobenius-Euer numbers and poynomias T. Kim, Barnes-type mutipe -zeta functions and -Euer poynomias, J. Phys. A 43 (200, no. 25, 25520, pp. 0. B. A, Kupershmidt, Refection symmetries of -Bernoui poynomias, J. Noninear Math. Phys. 2 (2005, supp., N. I. Mahmudov, On a cass of -Bernoui and -Euer poynomias, Adv. Difference Eu. 203, 203: D.S. Kim, T. Kim, Y. H. Kim, S. H. Lee, Some arithmetic properties of Bernoui and Euer numbers, Adv. Stud. Contemp. Math. 22 (202, no. 4, S. Roman, The umbra cacuus, Pure and Appied Mathematics,.Academic Press, Inc. [Harcourt Brace Jovanovich, Pubishers], New Yor, 984. x+93 pp. ISBN: S. Roman, G.-C. Rota, The umbra cacuus, Advances in Math. 27 (978, no. 2, Received: September, 203