The adjunction property, recently introduced for Sheffer polynomial sets, is considered in the case of Appell polynomials. The particular case of adjoint Appell-Euler and Appell-Bernoulli polynomials of the first kind is analyzed

Natalini, Pierpaolo

Ricci, Paolo E.

English

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Applications and Applied Mathematics: An International Journal (AAM) Volume 14 Issue 2 Article 31 12-2019 Adjoint Appell-Euler and First Kind Appell-Bernoulli Polynomials Pierpaolo Natalini Università degli Studi Roma Tre Paolo E. Ricci International Telematic University UniNettuno Follow this and additional works at: https://digitalcommons.pvamu.edu/aam  Part of the Discrete Mathematics and Combinatorics Commons, Number Theory Commons, and the Special Functions Commons Recommended Citation Natalini, Pierpaolo and Ricci, Paolo E. (2019). Adjoint Appell-Euler and First Kind Appell-Bernoulli Polynomials, Applications and Applied Mathematics: An International Journal (AAM), Vol. 14, Iss. 2, Article 31. Available at: https://digitalcommons.pvamu.edu/aam/vol14/iss2/31 This Article is brought to you for free and open access by Digital Commons @PVAMU. It has been accepted for inclusion in Applications and Applied Mathematics: An International Journal (AAM) by an authorized editor of Digital Commons @PVAMU. For more information, please contact hvkoshy@pvamu.edu. Available athttp://pvamu.edu/aamAppl. Appl. Math.ISSN: 1932-9466Applications and AppliedMathematics:An International Journal(AAM)Vol. 14, Issue 2 (December 2019), pp. 1112 – 1122Adjoint Appell-Euler and First Kind Appell-Bernoulli Polynomials1Pierpaolo Natalini and 2Paolo E. Ricci1 Dipartimento di Matematica e FisicaUniversità degli Studi Roma TreLargo San Leonardo Murialdo, 1Roma - 00146, Italianatalini@mat.uniroma3.it2 International Telematic University UniNettunoCorso Vittorio Emanuele II, 39Roma - 00186, Italiapaoloemilioricci@gmail.comReceived: September 9, 2018; Accepted: June 5, 2019AbstractThe adjunction property, recently introduced for Sheffer polynomial sets, is considered in the caseof Appell polynomials. The particular case of adjoint Appell-Euler and Appell-Bernoulli polyno-mials of the first kind is analyzed.Keywords: Appell polynomials; Generating functions; Monomiality principle; Differentialequations; Combinatorial analysisMSC 2010 No.: 33C99, 05A10, 11P811. IntroductionIn recent articles (refer to Bretti et al. (2018a), Ricci et al. (2017)), new sets of Sheffer (referto Sheffer (1939)) and Brenke (refer to Brenke (1945)) polynomials, based on higher order Bellnumbers (refer to Bernardini et al. (2005), Natalini and Ricci (2004), Natalini and Ricci (2016),11121Natalini and Ricci: Adjoint Appell-Euler and First Kind Appell-Bernoulli PolynomialsPublished by Digital Commons @PVAMU, 2019AAM: Intern. J., Vol. 14, Issue 2 (December 2019) 1113Natalini and Ricci (2018), Ricci et al. (2017)), have been studied. Furthermore, the notion of adjointSheffer polynomial sets has been introduced (refer to Ricci (2018)) and applied in several cases(refer to Bretti et al. (2018b), Natalini et al. (2018), Ricci et al. (2019)). In this article we considera similar notion in the case of Appell polynomial sets, starting by the case of Euler and first kindBernoulli polynomials.2. Sheffer polynomialsThe Sheffer polynomials {sn(x)} are introduced (refer to Sheffer (1939)) by means of the expo-nential generating function (refer to Srivastava and Manocha (1984)) of the type:A(t) exp(xH(t)) =∞∑n=0sn(x)tnn!, (1)whereA(t) =∞∑n=0antnn!, (a0 6= 0) ,H(t) =∞∑n=0hntnn!, (h0 = 0) .(2)According to a different characterization (refer to Roman (1984), [p. 18]), the same polynomialsequence can be defined by means of the pair (g(t), f(t)), where g(t) is an invertible series andf(t) is a delta series:g(t) =∞∑n=0gntnn!, (g0 6= 0) ,f(t) =∞∑n=0fntnn!, (f0 = 0, f1 6= 0) .(3)Denoting by f−1(t) the compositional inverse of f(t) (i.e. such that f (f−1(t)) = f−1 (f(t)) = t),the exponential generating function of the sequence {sn(x)} is given by1g[f−1(t)]exp(xf−1(t))=∞∑n=0sn(x)tnn!, (4)so thatA(t) =1g[f−1(t)], H(t) = f−1(t) . (5)When g(t) ≡ 1, the Sheffer sequence corresponding to the pair (1, f(t)) is called the associatedSheffer sequence {σn(x)} for f(t), and its exponential generating function is given byexp(xf−1(t))=∞∑n=0σn(x)tnn!. (6)2Applications and Applied Mathematics: An International Journal (AAM), Vol. 14 [2019], Iss. 2, Art. 31https://digitalcommons.pvamu.edu/aam/vol14/iss2/311114 P. Natalini and P.E. RicciA list of known Sheffer polynomial sequences and their associated ones can be found in Boas andBuck (1958b).3. Adjointness for Sheffer polynomial setsAccording to the above considerations, Sheffer polynomials are characterized both by the orderedcouples (A(t), H(t)), or by (g(t), f(t)).Definition 3.1.Adjoint Sheffer polynomials are defined by interchanging the ordered couple (A(t), H(t)) with(g(t), f(t)), when writing the generating function.Here and in the following, the tilde “∼” above the symbol of a polynomial set stands for theadjective “adjoint”.3.1. Adjointness for Appell polynomial setsIn the case of Appell polynomials, since H(t) ≡ t, so that f(t) = H−1(t) = t, and A(t) = 1/g(t),the above definition becomes:Definition 3.2.Adjoint Appell polynomials are simply defined by changing A(t) with 1/A(t) in the generatingfunction.The simplest examples are given by the Adjoint Appell-Euler and the Adjoint Appell-Bernoullipolynomials of the first kind:1. Adjoint Appell-Euler polynomialsA(t) =et + 12,G(t, x) =et + 12ext =∞∑n=0ε̃n(x)tnn!.(7)2. Adjoint Appell-Bernoulli polynomials of the first kindA(t) =et − 1t,G(t, x) =et − 1text =∞∑n=0β̃n(x)tnn!.(8)3Natalini and Ricci: Adjoint Appell-Euler and First Kind Appell-Bernoulli PolynomialsPublished by Digital Commons @PVAMU, 2019AAM: Intern. J., Vol. 14, Issue 2 (December 2019) 11154. Adjoint Appell-Euler polynomialsWe recall that a polynomial set {pn(x)} is called quasi-monomial if and only if there exist twooperators P̂ and M̂ such thatP̂ (pn(x)) = n pn−1(x) ,M̂ (pn(x)) = pn+1(x) , (n = 1, 2, . . . ).(9)P̂ is called the derivative operator and M̂ the multiplication operator, as they act in the same wayof classical operators on monomials.This definition traces back to a paper by Steffensen (1941), recently improved by Dattoli (2000)and widely used in several applications (refer to Dattoli et al. (2003), Dattoli et al. (2009), and thereferences therein).Since we are considering an Appell polynomial set, the derivative operator is simply given by thederivative with respect to x, that isP̂ = Dx , Dx ε̃n(x) = n ε̃n−1(x) . (10)This can be easily checked by partially differentiating the generating function (7) with respect tox.The relevant differential equation is given by∂G∂x= tet − 12ext . (11)4.1. Recurrence relationTheorem 4.1.For any k ≥ 0, the polynomials ε̃k(x) satisfy the recurrence relation:ε̃k+1(x) = (1 + x) ε̃k(x)−xk2. (12)Proof:Differentiating equation (7) with respect to t we find:∂G∂t=∞∑k=0ε̃k+1(x)tkk!= (1 + x)G(t, x)− ext2= (1 + x)∞∑k=0ε̃k(x)tkk!− 12∞∑k=0xktkk!, (13)so that the recurrence (12) follows. 4Applications and Applied Mathematics: An International Journal (AAM), Vol. 14 [2019], Iss. 2, Art. 31https://digitalcommons.pvamu.edu/aam/vol14/iss2/311116 P. Natalini and P.E. Ricci4.2. Generating function’s PDETheorem 4.2.The generating function (7) satisfies the linear PDEt∂G∂t= (1 + x)∂G∂x− 1et + 1∂G∂x. (14)Proof:It is sufficient to eliminate the exponential function between the partial derivatives given in (11)and (13). Note that the last addend becomes negligible when t increases.4.3. Differential equationWe start noting thatA′(t)A(t)=etet + 1=∞∑k=1(−1)k+1tk . (15)Now, we can apply a general result by Ben Cheikh (2003) (Corollary 3.2), in order to find themultiplication operator for the quasi-monomial set {ε̃k(x)}, which at present is given byM̂ =A′(Dx)A(Dx)+ x =∞∑k=1(−1)k+1Dkx + x , (16)According to the results of monomiality principle (refer to Dattoli (2000)), the quasi-monomialpolynomials {ε̃n(x)} satisfy the differential equationM̂P̂ ε̃n(x) = n ε̃n(x) . (17)In the present case, we haveM̂P̂ =[∞∑k=1(−1)k+1Dkx + x]Dx , (18)so that we have the following theorem.Theorem 4.3.The adjoint Appell-Euler polynomials {ε̃n(x)} satisfy the differential equation[n∑k=2(−1)kDkx + xDx]ε̃n(x) = n ε̃n(x) , (19)5Natalini and Ricci: Adjoint Appell-Euler and First Kind Appell-Bernoulli PolynomialsPublished by Digital Commons @PVAMU, 2019AAM: Intern. J., Vol. 14, Issue 2 (December 2019) 1117because, for any fixed n, the series expansion in Equation (18) reduces to a finite sum when it isapplied to a polynomial of degree n.Remark 4.4.The first few values of the adjoint Appell-Euler polynomials are as follows:ε̃0(x) = 1,ε̃1(x) = x+12,ε̃2(x) = x2 + x+ 12,ε̃3(x) = x3 + 32x2 + 32x+ 12,ε̃4(x) = x4 + 2x3 + 3x2 + 2x+ 12,ε̃5(x) = x5 + 52x4 + 5x3 + 5x2 ++52x+ 12,ε̃6(x) = x6 + 3x5 + 152x4 + 10x3 + 152x2 + 3x+ 12,ε̃7(x) = x7 + 72x6 + 212x5 + 352x4 + 352x3 + 212x2 + 72x+ 12,ε̃8(x) = x8 + 4x7 + 14x6 + 28x5 + 35x4 + 28x3 + 14x2 + 4x+ 12.Remark 4.5.Note that the above scheme satisfy a particular symmetry. Writing the polynomials in the formε̃n(x) = cn,0xn + cn,1xn−1 + · · ·+ cn,n−1x+ cn,n , (n ≥ 0) ,the coefficients cn,h , (0 ≤ h ≤ n) verify the conditions:cn,0 = 1 , ∀ n ≥ 0,cn,n =12, ∀ n ≥ 1,ch,k = ch,h−k , ∀ 1 ≤ h ≤ n− 1 , 1 ≤ k ≤ n− 1 .5. Adjoint Appell-Bernoulli polynomials of the first kindEven in this case, the derivative operator is simply given by the derivative with respect to x, that isP̂ = Dx , Dx β̃n(x) = n β̃n−1(x) , (20)as it can be easily checked by differentiating the generating function (8) with respect to x.The relevant differential equation is given by∂G∂x= (et − 1) ext . (21)6Applications and Applied Mathematics: An International Journal (AAM), Vol. 14 [2019], Iss. 2, Art. 31https://digitalcommons.pvamu.edu/aam/vol14/iss2/311118 P. Natalini and P.E. Ricci5.1. Recurrence relationTheorem 5.1.For any k ≥ 0, the polynomials ε̃k(x) satisfy the recurrence relation:(k + 1)β̃k(x) = (1 + x) k β̃k−1(x) + xk . (22)Proof:Differentiating Equation (8) with respect to t we find:∂G∂t=(tet − et + 1t2+ xet − 1t)ext ,and consequentlyt∂G∂t= (1 + x) tG(t, x)−G(t, x) + ext ,that is∞∑k=0kβ̃k(x)tkk!= (1 + x)∞∑k=0k β̃k−1(x)tkk!−∞∑k=0β̃k(x)tkk!+∞∑k=0xktkk!,so that the recurrence (21) follows. 5.2. Generating function’s PDETheorem 5.2.The generating function (7) satisfies the linear PDEt∂G∂t=[(1 + x)− 1t]∂G∂x+1et − 1∂G∂x, (t > 0) . (23)Proof:It is sufficient to eliminate the exponential function between the partial derivatives stated in (11)and (13). Note that the last addend becomes negligible when t increases.5.3. Differential equationFirst, note that the following expansion holds:tet − et + 1t(et − 1)= a0 + a1t+ a2t2 + a3t3 + . . . , (24)7Natalini and Ricci: Adjoint Appell-Euler and First Kind Appell-Bernoulli PolynomialsPublished by Digital Commons @PVAMU, 2019AAM: Intern. J., Vol. 14, Issue 2 (December 2019) 1119where the coefficients ak are solution of the following triangular systema0 =12,a1 =13− a02!,a2 =18− a12!− a03!,. . .ak =k+1(k+2)!−k−1∑`=0a`(k + 1− `)!,(25)so that, for the first few values, we find:a0 =12, a1 =112,a2 = 0, a3 = − 1720 ,a4 = a2h = 0 , ∀h ≥ 1a5 =130240, a7 = − 11209600 ,a9 =1479001160, a11 = − 11307674368000 ,By applying the same result by Ben Cheikh (2003) (Corollary 3.2), and using expansion (24), wefind that the multiplication operator for the quasi-monomial set{β̃k(x)}is given byM̂ =A′(Dx)A(Dx)+ x =∞∑k=0akDkx + x . (26)Therefore, we have the result:Theorem 5.3.The adjoint Appell-Bernoulli polynomials of the first kind {β̃n(x)} satisfy the differential equa-tionM̂P̂ β̃n(x) =(∞∑k=0akDk+1x + xDx)β̃n(x) = n β̃n(x) ,that is (n−1∑k=0akDk+1x + xDx)β̃n(x) = n β̃n(x) , (27)because, for any fixed n, the series expansion in Equation (26) reduces to a finite sum when it isapplied to a polynomial of degree n.8Applications and Applied Mathematics: An International Journal (AAM), Vol. 14 [2019], Iss. 2, Art. 31https://digitalcommons.pvamu.edu/aam/vol14/iss2/311120 P. Natalini and P.E. RicciRemark 5.4.The first few values of the adjoint Appell-Bernoulli polynomials of the first kind are as follows:β̃0(x) = 1,β̃1(x) = x+12,β̃2(x) = x2 + x+ 13,β̃3(x) = x3 + 32x2 + x+ 14,β̃4(x) = x4 + 2x3 + 2x2 + x+ 15,β̃5(x) = x5 + 52x4 + 103x3 + 52x2 + x+ 16,β̃6(x) = x6 + 3x5 + 5x4 + 5x3 + 3x2 + x+ 17,β̃7(x) = x7 + 72x6 + 7x5 + 354x4 + 7x3 + 72x2 + x+ 18,β̃8(x) = x8 + 4x7 + 283x6 + 14x5 + 14x4 + 283x3 + 4x2 + x+ 19.Remark 5.5.Note that the above scheme satisfy a particular symmetry. Writing the polynomials in the formβ̃n(x) = dn,0xn + dn,1xn−1 + · · ·+ dn,n−1x+ dn,n , (n ≥ 0) ,the coefficients dn,h , (0 ≤ h ≤ n) verify the conditions:dn,0 = 1 , ∀ n ≥ 0,dn,n−1 = 1 , ∀ n ≥ 2,dn,n =1n+1, ∀ n ≥ 1,dh,k = dh,h−k−1 , ∀ 1 ≤ h ≤ n− 1 , 1 ≤ k ≤ n− 1 , (n ≥ 4) .6. ConclusionWe have introduced two set of adjoint Appell polynomials, called adjoint Euler and adjointBernoulli polynomials, in the framework of a general technique associating a new Appell poly-nomial set starting from a given one. We have derived their main characteristics by using themonomiality property and we have found their shift operators by means of a preceding result by Y.Ben Cheihk. Therefore, the differential equation satisfied by these polynomials have been shown.It is worth to note that the relevant differential equations are of infinite order, but actually theyreduce to finite order when applied to each polynomial.For the classical Euler and Bernoulli polynomials the same result was proved in He and Ricci(2002), where a similar phenomenon was found for the recursion satisfied by the classical Euler andBernoulli polynomials. In fact, it was noticed that the order of these recurrence relations increaseswith polynomial degree.9Natalini and Ricci: Adjoint Appell-Euler and First Kind Appell-Bernoulli PolynomialsPublished by Digital Commons @PVAMU, 2019AAM: Intern. J., Vol. 14, Issue 2 (December 2019) 1121REFERENCESBell, E.T. (1934). 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Adjoint Appell-Euler and First Kind Appell-Bernoulli Polynomials

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Adjoint Appell-Euler and First Kind Appell-Bernoulli Polynomials

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