J. Ana. Num. Theor. 6, No. 2, 51-55 (2018) 51 Journal of Analysis & Number Theory An International Journal

http://dx.doi.org/10.18576/jant/060203

New Sets of Euler-Type

1 2, 3 Gabriella Bretti , Pierpaolo Natalini ∗ and Paolo Emilio Ricci 1 Istituto per le Applicazioni del Calcolo (IAC) “M. Picone” Consiglio Nazionale delle Ricerche (CNR - National Research Council) Via dei Taurini, Roma, Italia 2 Dipartimento di Matematica e Fisica-Universit`adegli Studi Roma Tre Largo San Leonardo Murialdo, Roma, Italia 3 International Telematic University UniNettuno, Corso Vittorio Emanuele II, 39, 00186 - Roma, Italia

Received: 2 May 2018, Revised: 23 Jun. 2018, Accepted: 28 Jun. 2018 Published online: 1 Jul. 2018

Abstract: In recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers have been studied, and several integer sequences related to them have been introduced. In the article other types of Sheffer polynomials are considered, by introducing two sets of Euler-type polynomials.

Keywords: Sheffer polynomials, combinatorial analysis, Bernoulli numbers, Euler-type polynomials

1 Introduction the pair (g(t), f (t)), where g(t) is an invertible series and f (t) is a delta series: In recent articles [5,12], the present authors have studied n ∑∞ t new sets of Sheffer [14] and Brenke [4] polynomials g(t)= n=0 gn n! , (g0 = 0), 6 (3) related to higher order Bell numbers. Furthermore, ∞ tn f (t)= ∑ fn , ( f0 = 0, f1 = 0). several integer sequences associated with the considered n=0 n! 6 1 polynomials sequences, both of exponential and Denoting by f − (t) the compositional inverse of f (t) (i.e. 1 1 logarithmic type, have been highlighted. such that f f − (t) = f − ( f (t)) = t), the exponential It is worth to note that exponential and logarithmic  of the sequence sn(x) is given by polynomials have been recently studied in the ∞ {n } 1 1 t multivariate case [9,10,11]. exp x f − (t) = ∑ sn(x) , (4) g[ f 1(t)] n! In this article we show new sets of Sheffer polynomials, −  n=0 by introducing two families of Euler-type polynomials. so that 1 1 A(t)= 1 , H(t)= f − (t). (5) g[ f − (t)] 2 Sheffer polynomials When g(t) 1, the Sheffer sequence corresponding to the pair (1, f (t≡)) is called the associated Sheffer sequence The Sheffer polynomials s (x) are introduced [14] by σn(x) for f (t), and its exponential generating function n { } means of the exponential{ generating} function [15] of the is given by type: ∞ tn ∞ 1 σ tn exp x f − (t) = ∑ n(x) . (6) A(t)exp(xH(t)) = ∑ s (x) , (1) n=0 n! n n!  n=0 A list of known Sheffer sequences and their where associated ones can be found in [2,3]. ∑∞ tn A(t)= n=0 an n! , (a0 = 0), 6 (2) ∑∞ tn 3 Euler-type polynomials H(t)= n=0 hn n! , (h0 = 0). According to a different characterization (see [13, p. 18]), Here we introduce a Sheffer polynomial set connected the same can be defined by means of with the classical Euler polynomials.

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Assuming: Proof. - Differentiating G(t,x) with respect to t , we have ∂ 1 G(t,x) sinht G t x A t H t sinht (1) ∂t = cosht ( , ) ( )= , ( )= , − (9) cosht ∞ n 1 ∑ ˜ t − +x cosht G(t,x)= n=1 En(x) (n 1)! . we consider the Euler-type polynomials E˜n(x), defined by − the generating function that is, ∞ n k t 1 ∑∞ ˜ t (2) ˜ G(t,x)= cosht exp[x sinht]= k=0 Ek(x) k! . x cosht G(t,x) tanht G(t,x)= ∑ En+1(x) . − n=0 n! Note that the Euler numbers are recovered, since we have: Putting ∞ k 2k 2k 2 t ∞ 2 (2 1)B2k G(t,0)= = ∑ E˜ (0) , (3) ∑ − 2k 1 t t k tanht = k=1 (2k)! t − e + e− k=0 k! k+1 k+1 k+1 ∞ 1+( 1) 2 (2 1)Bk+1 k ∞ k ∑ − t ∑ τ t so that E˜n(0)= En. = k=0 −2 k+1 k! = k=0 k k! , In what follows, we use the expansions   we find 2k+1 k+1 k k ∑∞ t ∑∞ 1+( 1) t (4) ∑∞ 1+( 1) tk ∑∞ ˜ tk sinht = k=0 (2k+1)! = k=0 −2 k! , x k=0 2− k! k=0 Ek(x) k!     ∞ tk ∞ ∞ tk k ∑ τk ∑ E˜k(x)= ∑ E˜k 1(x) , ∑∞ t2k ∑∞ 1+( 1) tk − k=0 k! k=0 n=0 + k! cosht = k=0 (2k)! = k=0 2− k! . (5)   and therefore k k h k ∑∞ ˜ t ∑∞ ∑k k 1+( 1) − ˜ t Theorem 1.- For any k 0, the polynomials E˜k(x) satisfy k=0 Ek+1(x) k! = k=0 h=0 h −2 xEh(x) k! the differential identity:≥   ∞ k k tk ∑ ∑ τk hE˜h(x) , k h+1 k=0 h=0 h k! ˜ ∑k k 1+( 1) − ˜ − − Ek′ (x)= h=0 h − 2 Eh(x). (6) so that the recurrence relation  (8) follows.    We recall that a polynomial set pn(x) is called quasi- Proof. - We write equation (2) in the form monomial if and only if there exist{ two operators} Pˆ and Mˆ ∞ k such that ˜ t exp[x sinht]= cosht ∑ Ek(x) , Pˆ (pn(x)) = npn 1(x) k=0 k! − (10) Mˆ (p (x)) = p (x), (n = 1,2,...). by differentiating with respect to x, we find n n+1 Pˆ is called the derivative operator and Mˆ the multiplication ∑∞ ˜ tk sinht exp[x sinht]= cosht k=1 Ek′ (x) k! , operator, as they act in the same way of classical operators on monomials. i.e., This definition traces back to a paper by J.F. Steffensen ∑∞ ˜ tk ∑∞ ˜ tk [16], recently improved by G. Dattoli [6] and widely used sinht G(t,x)= sinht k=0 Ek(x) k! = k=1 Ek′ (x) k! . in several applications (see e.g. [7,8] and the references Therefore, by using equation (4), we have therein). ∞ Y. Ben Cheikh [1] proved that every polynomial set is sinht G(t,x)= ∑ ∑k k k=0 h=0 h quasi-monomial under the action of suitable derivative k h+1 k  k (7) 1+( 1) − ˜ t ∑∞ ˜ t and multiplication operators. In particular, in the same − 2 Eh(x) k! = k=1 Ek′ (x) k! .   article (Corollary 3.2), the following result is proved ˜ and putting E0′ (x)= 0, equation (6) follows. Theorem 3.- Let (pn(x)) denote a Boas-Buck polynomial set, i.e. a set defined by the generating function Theorem 2.- For any k 0, the polynomials E˜k(x) satisfy ≥ ∞ tn the following recurrence relation: A(t)ψ(xH(t)) = ∑ p (x) , (11) n n! k h n=0 ˜ ∑k k 1+( 1) − τ ˜ Ek+1(x)= h=0 h −2 x k h Eh(x), (8) h  − − i where  ∞ n where A(t)= ∑ a˜nt , (a˜0 = 0), n=0 6 (12) k+1 k+1 k+1 1 + ( 1) 2 2 1 Bk+1 ∞ n τ = − − ψ(t)= ∑ γ˜nt , (γ˜n = 0 n), k  2  k + 1  n=0 6 ∀ with ψ(t) not a polynomial, and lastly are the coefficients of the McLaurin expansion of the ∞ n+1 H(t)= ∑ h˜n t , (h˜0 = 0). (13) function tanht,andBk are the Bernoulli numbers. n=0 6

c 2018 NSP Natural Sciences Publishing Cor. J. Ana. Num. Theor. 6, No. 2, 51-55 (2018) / www.naturalspublishing.com/Journals.asp 53

( ) Let σ Λ the lowering operator defined by Theorem 5.- The Euler-type polynomials E˜n(x) satisfy ∈ − { } γ the differential equation n ˜n 1 n 1 σ(1)= 0, σ(x )= − x − , (n = 1,2,...)(14). γ˜ 2 ∞ 1/2 2k n (xD Dx + x)∑ − D x − k=0 k x Put nh  i (20) ∑∞ k (2k)! 2k+1 ˜ ˜ γ˜ k=0( 1) k 2 Dx En(x)= nEn(x), σ 1 n n+1 n+1 − 4 (k!) (2k+1) − (x )= γ x (n = 0,1,2,...). (15) o ˜n i.e. n 1 Denoting, as before, by f (t) the compositional inverse of 2 [ −2 ] k h (xD Dx + x)∑ ∑ ( 1) H(t), the Boas-Buck polynomial set pn(x) is x k=0 h=0 { } − − (21) quasi-monomial under the action of the operators 1/2 (2h)! D2k+1 E˜ x nE˜ x −k h 4h(h!)2(2h+1) x n( )= n( ). σ − A′[ f ( )] 1  Pˆ = f (σ), Mˆ = + xDxH′[ f (σ)]σ − , (16) Note that, for any fixed n, the Cauchy product of series A[ f (σ)] expansions in equation (20) reduces to a finite sum, with n 1 where prime denotes the ordinary derivatives with respect upper limit −2 , when it is applied to a polynomial of to t. degree n, because  the successive addends vanish. Note that in our case we are dealing with a Sheffer Remark.The first few Euler-type polynomials are as polynomial set, so that since we have ψ(t) = et , the follows: operator σ defined by equation (14) simply reduces to the E˜ (x)= 1 derivative operator Dx. Furthermore, we have: 0 E˜1(x)= x 1 A′(t) 2 A(t) = cosht , A(t) = tanht , E˜2(x)= x 1 − 3 − E˜3(x)= x 2x 4 − 2 ∞ t2k+1 1+( 1)k+1 1 E˜4(x)= x 2x + 5 H(t) = sinht = ∑ , h˜ = − , k=0 (2k+1)! k 2 (2k+1)! ˜ 5 −     E5(x)= x + 16x 6 4 2 E˜6(x)= x + 5x + 31x 61 1 2 ˜ 7 5 3− H′(t) = cosht , f (t) = H− (t) = log(t + √t + 1), E7(x)= x + 14x + 56x 272x 8 6 4− 2 E˜8(x)= x + 28x + 126x 692x + 1385 so that we have the theorem 9 7 5 − 3 E˜9(x)= x + 48x + 336x 1280x + 7936x ˜ 10 8 −6 4 2 Theorem 4.- The Euler-type polynomial set En(x) is E˜10(x)= x + 75x + 882x 1490x + 25261x 50521 quasi-monomial under the action of the operators{ } − − ˆ 2 P = log(Dx + Dx + 1) 4 Circular Euler-type polynomials p (17) Mˆ = tanh(settsinhDx)+ xsettsinhDx , − Assuming: (by settsinht = log(t + √t2 + 1) we denote the inverse of 1 the function sinht), A(t)= , H(t)= sint , (1) i.e. cost ∞ ˜ Pˆ = ∑ ( 1)k (2k)! D2k+1 , we consider the circular Euler-type polynomials Sn(x), k=0 − 4k(k!)2(2k+1) x defined by the generating function 1 ∑∞ ˜ tk (2) G(t,x)= cost exp[x sint]= k=0 Sk(x) k! . ˆ Dx 2 M = 2 + x 1 + Dx − √1+Dx p (18) Remark.The first few circular Euler-type polynomials are 2 2 1/2 =(xD Dx + x)(1 + D ) , as follows: x − x − S˜0(x)= 1 ˜ ˆ 2 ∑∞ 1/2 2k S1(x)= x M =(xDx Dx + x) k=0 − k Dx . 2 − S˜2(x)= x + 1  3 There is no problem about the convergence of the above S˜3(x)= x + 2x 4 2 series, since they reduce to finite sums when applied to S˜4(x)= x + 2x + 5 polynomials. 5 S˜5(x)= x + 16x According to the results of monomiality principle [6], 6 4 2 S˜6(x)= x 5x + 31x + 61 the quasi-monomial polynomials pn(x) satisfy the ˜ 7 − 5 3 { } S7(x)= x 14x + 56x + 272x. differential equation 8 − 6 4 2 S˜8(x)= x 28x + 126x + 692x + 1385 ˆ ˆ 9 − 7 5 3 MPpn(x)= npn(x). (19) S˜9(x)= x 48x + 336x + 1280x + 7936x 10− 8 6 4 2 S˜10(x)= x 75x + 882x + 1490x + 25261x + 50521 In the present case, we have −

c 2018 NSP Natural Sciences Publishing Cor. 54 G. Bretti et al.: New sets of Euler-type polynomials

Remark.Therefore, we can write a table of circular Euler- [10] F. Qi, Integral representations for multivariate type numbers: logarithmic potentials, J. Comput. Appl. Math. 336 (2018), 54–62; available on line at S˜0(0)= 1 S˜2(0)= 1 https://doi.org/10.1016/j.cam.2017.11.047. S˜4(0)= 5 S˜6(0)= 61 [11] F. Qi, On multivariate logarithmic polynomials and their S˜8(0)= 1385 S˜10(0)= 50521 properties, Indag. Math. 336 (2018), (in press); available on line at https://doi.org/10.1016/j.indag.2018.04.002. S˜2k+1(0)= 0, k 0. [12] P.E. Ricci, P. Natalini, G. Bretti, Sheffer and ∀ ≥ Brenke polynomials associated with generalized Note that the Euler numbers (in absolute value) appear, Bell numbers, Jnanabha, Vijnana Parishad of since we have: India, 47 (No2-2017), 337–352; available on line at http://www.vijnanaparishadofindia.org/jnanabha/volume-47- S˜2k(0)= E˜2k . (3) | | no-2-2017 Taking into account the relations between the hyperbolic [13] S.M. Roman, The , Academic Press, New and circular functions: York, 1984. [14] I.M. Sheffer, Some properties of polynomials sets of zero sinh(ix)= i sinx, cosh(ix)= cosx, type, Duke Math. J., 5 (1939), 590–622; available online at https://projecteuclid.org/euclid.dmj/1077491414 the following relation between the polynomials S˜k(x) and ˜ [15] H.M. Srivastava, H.L. Manocha, A Treatise on Generating Ek(x) follows: Functions, Halsted Press (Ellis Horwood Limited, k Chichester), John Wiley and Sons, New York, Chichester, S˜k(ix)= i E˜k(x). (4) Brisbane and Toronto, 1984. [16] J.F. Steffensen, The poweroid, an extension of the mathematical notion of power, Acta Math., 73 (1941), References 333–366; available online at doi:10.1007/BF02392231, https://projecteuclid.org/euclid.acta/1485888329. [1] Y.B. Cheikh, Some results on quasi-monomiality, Appl. Math. Comput., 141 (2003), 63–76. [2] R.P. Boas, R.C. Buck, Polynomials defined by generating relations, Amer. Math. Monthly, 63 (1958), 626–632; G. Bretti is a researcher available online at https://www.jstor.org/stable/2310587. at the National Researh [3] R.P. Boas, R.C. Buck, Polynomial Expansions of Analytic Council (I.A.C.) in Rome, Functions, Springer-Verlag, Berlin, Gottingen, Heidelberg, Italy. She works by many New York, 1958. years in Numerical Analysis, [4] W.C. Brenke, On generating functions of polynomial Special functions, Matrix systems, Amer. Math. Monthly, 52 (1945), 297–301; available functions, Integral and online at https://www.jstor.org/stable/2305289. discrete transforms, Ordinary [5] G. Bretti, P. Natalini, P.E. Ricci, A new set of Sheffer-Bell and partial differential polynomials and logarithmic numbers, Georgian Math. J., equations, Asymptotic 2018, (to appear). analysis, Eigenvalue problems. [6] G. Dattoli, Hermite-Bessel and Laguerre-Bessel functions: A by-product of the monomiality principle, in Advanced Special Functions and Applications (Proceedings of the Melfi School on Advanced Topics in Mathematics and Physics; Melfi, 9-12 May, 1999) (D. Cocolicchio, G. Dattoli and H.M. P. Natalini is professor Srivastava, Editors), Aracne Editrice, Rome, 2000, pp. 147– and researcher at the 164. RomaTre University. [7] G. Dattoli, P.E. Ricci, H.M. Srivastava, Editors), Advanced He works by many years in Special Functions and Related Topics in Probability and in Numerical Analysis, Special Differential Equations, (Proceedings of the Melfi School on functions, Matrix functions, Advanced Topics in Mathematics and Physics; Melfi, June Integral and discrete 24-29, 2001), in: Applied Mathematics and Computation, transforms, Ordinary and 141, (No. 1) (2003), 1–230. partial differential equations, [8] G. Dattoli, B. Germano, M.R. Martinelli and P.E. Ricci, Asymptotic analysis, Monomiality and partial differential equations, Math. Eigenvalue problems. Comput. Modelling, 50 (2009), 1332–1337; available online at https://doi.org/10.1016/j.mcm.2009.06.013. [9] F. Qi, D.-W. Niu, D. Lim, and B.-N. Guo, Some properties and an application of multivariate exponential polynomials, HAL archives (2018); available online at https://hal.archives- ouvertes.fr/hal-01745173.

c 2018 NSP Natural Sciences Publishing Cor. J. Ana. Num. Theor. 6, No. 2, 51-55 (2018) / www.naturalspublishing.com/Journals.asp 55

P. E. Ricci is professor and researcher at the UniNettuno International Telematic University in Rome, being retired from Rome University Sapienza by more than nine years. He works by many years in Numerical Analysis, Special functions, Matrix functions, Integral and discrete transforms, Ordinary and partial differential equations, Asymptotic analysis, Eigenvalue problems.

c 2018 NSP Natural Sciences Publishing Cor.