Bazsó, András

Pink, István

Periodica Mathematica Hungarica

DEA University of Debrecen Electronic Archive

DIOPHANTINE EQUATIONS WITH APPELLSEQUENCESANDRA´S BAZSO´ AND ISTVA´N PINKAbstract. We consider the Diophantine equation Pn(x) = g(y)in x, y where Pn(x), g(x) ∈ Q[x],deg g(x) ≥ 3 and {Pn(x)}n≥0 isan Appell sequence. Under some reasonable assumptions on Pn(x)we prove an ineffective finiteness result on the above equation.1. IntroductionFor n ∈ N∪{0}, let Pn(x) be a polynomial with rational coefficientsand with degPn(x) = n. Further, let P0(x) be a non-zero constant.The sequence {Pn(x)}n≥0 is called an Appell sequence (and Pn(x) iscalled an Appell polynomial) ifP ′n(x) = nPn−1(x) for all n ∈ N. (1)The history of such polynomials goes back to Appell’s work [2] in 1880.There are several well-known examples of Appell sequences, such asthe Bernoulli polynomials Bn(x), the Euler polynomials En(x), andthe Hermite polynomials Hn(x), respectively defined by the followinggenerating series (see e.g. [12])t exp(tx)exp(t)− 1 =∞∑n=0Bn(x)tnn!;2 exp(xt)exp(t) + 1=∞∑n=0En(x)tnn!(|t| < pi);exp(tx)exp(t2/2)=∞∑n=0Hn(x)tnn!.The above defined Hermite polynomials Hn(x) are sometimes denotedby Hen(x), e.g. in the book of Abramowitz and Stegun [1].The following properties of Appell polynomials will often be used inthe text, sometimes without special reference.2010 Mathematics Subject Classification. 11D41, 11B83, 05A40.Key words and phrases. Diophantine equations, Appell sequences,decomposition.12 ANDRA´S BAZSO´ AND ISTVA´N PINKWe recall the so-called Appell Identity :Pn(x+ y) =n∑k=0(nk)Pk(x)yn−k =n∑k=0(nk)Pk(y)xn−k, (2)which, by setting y = 0, implies that there exists a sequence of rationalnumbers {cn}n≥0 with c0 6= 0 such thatPn(x) =n∑k=0(nk)ckxn−k, where ck := Pk(0) (k ≥ 0). (3)For the proofs of (2) and (3) see, for instance the book of Roman [12].Let K be an arbitrary field. We denote by K[x] the ring of polyno-mials in the variable x with coefficients from K. A decomposition of apolynomial F (x) over K is an equality of the following formF (x) = G1(G2(x)) (G1(x), G2(x) ∈ K[x]),which is nontrivial ifdegG1(x) > 1 and degG2(x) > 1.Two decompositions F (x) = G1(G2(x)) and F (x) = H1(H2(x)) aresaid to be equivalent if there exists a linear polynomial `(x) ∈ K[x]such that G1(x) = H1(`(x)) and H2(x) = `(G2(x)). The polynomialF (x) is called decomposable over K if it has at least one nontrivialdecomposition over K; otherwise it is said to be indecomposable.The decomposition of Bernoulli polynomials has been described byBilu et al. in [6]. Decomposition properties of Euler polynomials wererecently investigated by Rakaczki and Kreso [11]. The mentioned re-sults can both be summarized as follows: the corresponding polynomial(Bn(x) or En(x)) is indecomposable over C for all odd n, while, if nis even, then any nontrivial decomposition of the polynomial underconsideration over C is equivalent to one of the shapeP̂n/2((x− 12)2),where P̂n/2(x) is a polynomial of degree n/2 which is indecomposablefor every n. These results from [6] and [11] suggest the following notion.We say that an Appell sequence {Pn(x)}n≥0 is of special type if Pn(x) isindecomposable over C for all odd n, and, for even n, every nontrivialdecomposition of Pn(x) is equivalent to a decomposition of the formPn(x) = P̂n/2((x− 12)2), (4)DIOPHANTINE EQUATIONS WITH APPELL SEQUENCES 3with an indecomposable polynomial P̂n/2(x) over C of degree n/2. Cle-arly, the polynomials {Bn(x)}n≥0 and {En(x)}n≥0 are of special type.The theory of polynomial decomposition is strongly connected tothe theory of separable Diophantine equations since, in 2000, Biluand Tichy [5] established their general ineffective finiteness criterionon equations of the form f(x) = g(y). (See Proposition 1 below.)In this paper we study the Diophantine equationPn(x) = g(y) in integers x, y, (5)where Pn(x) is from an Appell sequence of special type and g(x) ∈ Q[x],deg g(x) ≥ 3. For technical reasons, we restrict ourselves to Appellsequences {Pn(x)}n≥0 for which3P2(−c1/c0)2 − 2c0P4(−c1/c0)3P2(−c1/c0)2 − c0P4(−c1/c0) is not a positive integer. (6)Remark. In the following table, we give the value of the constantfrom (6) for the case when Pn(x) is a Bernoulli, Euler or an Hermitepolynomial, respectively.Bn(x) En(x) Hn(x)9/2 7/2 undefinedFor Pn(x) = Bn(x), Rakaczki [10], and independently Kulkarni andSury [9] characterized those pairs (n, g(y)) for which equation (5) hasinfinitely many integer solutions. Recently, Rakaczki and Kreso [11]proved an analogous result for the case when Pn(x) = (En(0)±En(x))/2(which is not an Appell polynomial anymore). For further related re-sults we refer to [7, 8].We prove the following.Theorem 1. Let g(x) ∈ Q[x] with deg g(x) ≥ 3, and suppose that{Pn(x)}n≥0 is an Appell sequence of special type with property (6).Then for n ≥ 7, equation (5) has only finitely many integer solutionsx, y, apart from the following cases:(i) g(x) = Pn(h(x)), where h(x) is a polynomial over Q.(ii) g(x) = γ(δ(x)m), where m is a positive integer.(iii) n is even and g(x) = P̂n/2(q(x)2)(iv) n is even and g(x) = P̂n/2(δ(x)q(x)2)(v) n is even and g(x) = P̂n/2(cδ(x)t), where t ≥ 3 is an odd integer(vi) n is even and g(x) = P̂n/2((aδ(x)2 + b)q(x)2)Here a, b, c ∈ Q \ {0}, γ(x), δ(x) ∈ Q[x] are linear polynomials andq(x) ∈ Q[x] is a non-zero polynomial.4 ANDRA´S BAZSO´ AND ISTVA´N PINKWe prove the above theorem by applying among other things thegeneral finiteness criterion of Bilu and Tichy [5] for equation (5). Henceour finiteness result is ineffective.Remark. For n ≥ 7, our main result is a common generalizationof the mentioned results of Rakaczki [10], Kulkarni and Sury [9] andRakaczki and Kreso [11]. In the special cases Pn(x) ∈ {Bn(x), En(x)},one can exclude the exceptional case (ii) by making use of some specificproperties of the Bernoulli or Euler polynomials, respectively. (See[9, 10, 11])2. Auxiliary resultsBefore proving Theorem 1, we collect those results that we are goingto apply in the proof. First, we recall the finiteness criterion of Biluand Tichy [5]. To do this, we need to define five kinds of so-calledstandard pairs of polynomials.Let α, β be nonzero rational numbers, µ, ν, q > 0 and r ≥ 0 beintegers, and let v(x) ∈ Q[x] be a nonzero polynomial (which may beconstant). Denote by Dµ(x, δ) the µ-th Dickson polynomial, defined bythe functional equation Dµ(z + δ/z, δ) = zµ + (δ/z)µ or by the explicitformulaDµ(x, δ) =bµ/2c∑i=0dµ,ixµ−2i with dµ,i =µµ− i(µ− ii)(−δ)i.Two polynomials f1(x) and g1(x) are said to form a standard pair overQ if one of the ordered pairs (f1(x), g1(x)) or (g1(x), f1(x)) belongs tothe list below. The five kinds of standard pairs are then listed in thefollowing table.kind explicit form of {f1(x), g1(x)} parameter restrictionsfirst (xq, αxrv(x)q) 0 ≤ r < q, (r, q) = 1,r + deg v(x) > 0second (x2, (αx2 + β)v(x)2) -third (Dµ(x, αν), Dν(x, αµ)) (µ, ν) = 1fourth (α−µ2 Dµ(x, α),−β −ν2 Dν(x, β)) (µ, ν) = 2fifth ((αx2 − 1)3, 3x4 − 4x3) -The following proposition is a special case of the main result of [5].Proposition 1. Let f(x), g(x) ∈ Q[x] be nonconstant polynomialssuch that the equation f(x) = g(y) has infinitely many solutions inrational integers x, y. Then f = ϕ ◦ f1 ◦ λ and g = ϕ ◦ g1 ◦ µ,DIOPHANTINE EQUATIONS WITH APPELL SEQUENCES 5where λ(x), µ(x) ∈ Q[x] are linear polynomials, ϕ(x) ∈ Q[x], and(f1(x), g1(x)) is a standard pair over Q.For P (x) ∈ C[x], a complex number c is said to be an extremum ifP (x)− c has multiple roots. The P -type of c is defined to be the tuple(α1, . . . , αs) of the multiplicities of the distinct roots of P (x)− c in anincreasing order. Obviously, s < degP (x) and α1+. . .+αs = degP (x).Proposition 2. For a 6= 0 and k ≥ 3, Dµ(x, α) has exactly two extrema±2αµ2 . If µ is odd, then both are of P -type (1, 2, 2, . . . , 2). If µ is even,then 2αµ2 is of P -type (1, 1, 2, . . . , 2) and −2αµ2 is of P -type (2, 2, . . . , 2).Proof. See, for instance [4, Proposition 3.3]. We end this section with two technical results. Let d1, e1 ∈ Q∗ andd0, e0 ∈ QProposition 3. Suppose that {Pn(x)}n≥0 is an Appell sequence of spe-cial type. Then the polynomial Pn(d1x+d0) is not of the form e1xq+e0,with q ≥ 7.Proof. We assume the contrary, i.e., that we havePn(d1x+ d0) = e1xq + e0 (7)with q ≥ 7. Obviously, we then have n = q.We observe from (2) and (7) thatP1(d0) = P2(d0) = . . . = Pn−1(d0) = 0. (8)Since, by (3), P1(d0) = c0d0 + c1, we getd0 = −c1c0. (9)Further, since, by (1),Pk(x) =k!(n− 1)!P(n−1−k)n−1 (x), k = 1, . . . , n− 1, (10)we infer that d0 is a root of Pn−1(x) of multiplicity (n − 1). Thus, inview of (9), we have Pn−1(x) = c0 (x+ c1/c0)n−1, which impliesPn(x) = c0(x+c1c0)n+ C with C = Pn(−c1c0). (11)First, if n ≥ 7 is even , then, by (11), one can easily find the nont-rivial decomposition Pn(x) = Q(R(x)) withQ(x) = c0x2 + C, and R(x) =(x+c1c0)n/2. (12)6 ANDRA´S BAZSO´ AND ISTVA´N PINKSince n ≥ 7, this nontrivial decomposition is obviously not equivalentto the one in (4), contradicting that {Pn(x)}n≥0 is of special type.Now, let n ≥ 7 be an odd positive integer. If n is composite, thenany divisor v of n with 1 < v < n leads to a nontrivial decompositionPn(x) = c0((x+c1c0)v)n/v+ C, (13)which again contradicts that {Pn(x)}n≥0 is of special type (and in thiscase Pn(x) is indecomposable). If n is a prime, then derivating bothsides of (11) we obtainPn−1(x) = c0(x− 12)n−1, (14)where of course the exponent n − 1 is even. Similarly as above, thisleads to a nontrivial decomposition not equivalent to (4) and thus to acontradiction.Proposition 4. Suppose that {Pn(x)}n≥0 is an Appell sequence whichsatisfies (6). Then the polynomial Pn(d1x + d0) is not of the forme1Dµ(x, δ)+e0, where Dµ(x, δ) the µ-th Dickson polynomial with µ > 4,δ ∈ Q∗.Proof. Suppose that the Appell sequence {Pn(x)}n≥0 satisfies (6), andthat we havePn(d1x+ d0) = e1Dµ(x, δ) + e0. (15)Clearly, n = µ. Comparing the leading coefficients of both sides we getdn1c0 = e1, (16)where the numbers ck (k ≥ 0) are defined in (3). Similarly, from (2)and the equality of the coefficients of xn−1 on both sides we obtainndn−11 P1(d0) = 0, (17)which impliesd0 = −c1c0. (18)Again, by (2), comparing the coefficients of xn−2 gives(n2)dn−21 P2(d0) = −e1nδ, (19)whence, together with (16) it follows thatd21 = −(n− 1)P2(d0)2c0δ(20)DIOPHANTINE EQUATIONS WITH APPELL SEQUENCES 7Now we compare the coefficients of xn−4 on both sides of (15) and weobtain (n4)dn−41 P4(d0) =e1n(n− 3)δ22, (21)which along with (16) leads tod41 =(n− 1)(n− 2)P4(d0)12c0δ2. (22)After substituting (20) into (22), we obtain3(n− 1)P2(d0)2 = (n− 2)c0P4(d0), (23)whence, together with (18) it follows thatn =3P2(−c1/c0)2 − 2c0P4(−c1/c0)3P2(−c1/c0)2 − c0P4(−c1/c0) . (24)This is a contradiction by (6). We note that Proposition 4 is a common generalization of Lemma5.3 in [6], Lemma 2.4 in [3], and of the second statement of Lemma 12in [11].3. Proof of Theorem 1Let g(x) ∈ Q[x] with deg g(x) ≥ 3. Suppose that equation (5)has infinitely many integer solutions x, y with an Appell sequence{Pn(x)}n≥0 of special type satisfying (6) and with n ≥ 7. Thenby Proposition 1 it follows that there exist λ(x), µ(x), ϕ(x) ∈ Q[x],deg λ(x) = deg µ(x) = 1 such thatPn(x) = ϕ(f1(λ(x))) and g(x) = ϕ(g1(µ(x))), (25)where (f1(x), g1(x)) is a standard pair over Q.Let λ−1(x) = a1x + a0, µ−1(x) = b1x + b0, where a0, a1, b0, b1 ∈ Qwith a1b1 6= 0. Then we can rewrite (25) asPn(a1x+ a0) = ϕ(f1(x)) and g(b1x+ b0) = ϕ(g1(x)), (26)Since Pn(x) is of special type and degPn(x) = n, we obtain thatdegϕ(x) ∈{1,n2, n}.8 ANDRA´S BAZSO´ AND ISTVA´N PINK3.1. The case degϕ(x) = n. If we assume that degϕ(x) = n, then by(25), we have deg f1(x) = 1. Thus Pn(x) = ϕ(t(x)), where t(x) ∈ Q[x]is a linear polynomial. Clearly, t−1(x) ∈ Q[x] is also linear. By (25),we obtain Pn(t−1(x)) = ϕ(t(t−1(x))) = ϕ(x). Henceg(x) = ϕ(g1(µ(x))) = Pn(t−1(g1(µ(x)))) = Pn(q(x)), (27)where q(x) = t−1(g1(µ(x))). So, if, in our case, equation (5) has infin-itely many solutions, then g(x) is of the form as in Theorem 1 (i).3.2. The case degϕ(x) = 1. Let ϕ(x) = ϕ1x + ϕ0, where ϕ1, ϕ0 ∈ Qand ϕ1 6= 0. We study now the five kinds of standard pairs.In view of our assumptions on n and deg g(x), it follows that thestandard pair (f1(x), g1(x)) cannot be of the second or fifth kind.If it is of the third or fourth kind, we then have Pn(a1x + a0) =e1Dµ(x, δ) + e0 with e0 ∈ Q, e1, δ ∈ Q∗. This contradicts Proposition4.If (f1(x), g1(x)) is a standard pair of the first kind, then we haveeither(I) Pn(a1x+ a0) = ϕ1xq + ϕ0, or(II) Pn(a1x + a0) = ϕ1αxpν(x)q + ϕ0, where 0 ≤ p < q, (p, q) = 1and p+ deg ν(x) > 0.The first case (I) is impossible by Proposition 3 since n ≥ 7 byassumption.Let us now consider the second case (II). Then we have g(x) =ϕ1µ(x)q +ϕ0 = ϕ(µ(x)q), where q ≥ 3 and µ(x) ∈ Q[x] is linear, whichis case (ii) of Theorem 1.3.3. The case degϕ(x) = n/2. Clearly, n is then even, and from(25) we observe that deg f1(x) = 2. Hence it follows that, in (25),(f1(x), g1(x)) cannot be a standart pair of the fifth kind. Further,we obtain a nontrivial decomposition of Pn(x), which, since Pn(x) is ofspecial type, implies that there exists a linear polynomial `(x) = `1x+`0over Q such thatϕ(x) = P̂n/2(`(x)) and `(f1(λ(x))) =(x− 12)2. (28)Again, we study the remaining kinds of standard pairs.First, we consider the case when, in (25), (f1(x), g1(x))is a standardpair of the first kind. If f1(x) = xt, then by deg f1(x) = 2, we have(f1(x), g1(x)) = (x2, αxp(x)2). Putting λ(x) = λ1x + λ0, (28) takesthe form `((λ1x+λ0)2) = (x− 1/2)2, whence an easy calculation givesDIOPHANTINE EQUATIONS WITH APPELL SEQUENCES 9`(x) = x/λ21. Substituting this to (25), we obtaing(x) = P̂n/2(`(g1(µ(x)))) = P̂n/2(αµ(x)p(µ(x))2λ21)(29)So g(x) is of the form (iv) with δ(x) = αµ(x)/λ21 and q(x) = p(µ(x)).In the switched case (f1(x), g1(x)) = (αxrp(x)t, xt), where 0 ≤ r < t,(r, t) = 1 and r + deg p(x) > 0, deg f1(x) = 2 implies that one of thefollowing cases occurs:(A) r = 0, t = 1 and deg p(x) = 2, or(B) r = 2, t > 2 is odd and p(x) is a constant polynomial.In case (A) we have g1(x) = x, whence from (25) and (28) we obtaing(x) = P̂n/2(`(g1(µ(x)))) = P̂n/2(`(µ(x))) = P̂n/2(δ(x)q(x)2), (30)where δ(x) = `(µ(x)) and q(x) ≡ 1. Thus g(x) is again of the form(iv).In the second case (B), we can write f1(x) = βx2, with β = αp(x)t ∈Q \ {0}. Substituting this to (28), we deduce that `(x) = x/(βλ21),whence, by (25), we getg(x) = P̂n/2(`(g1(µ(x)))) = P̂n/2(µ(x)tβλ21)= P̂n/2(cδ(x)t), (31)where c = 1/(βλ21), δ(x) = µ(x) and t > 2 is odd. This is option (v)in Theorem 1.Next let, in (25), be a the standard pair (f1(x), g1(x)) of the secondkind. If (f1(x), g1(x)) = (x2, (αx2 + β)v(x)2), then a calculation from(28) yields `(x) = x/λ21, and by (25) we haveg(x) = P̂n/2(`(g1(µ(x)))) == P̂n/2((αµ(x)2 + β)v(µ(x))2λ21)= P̂n/2((αδ(x)2 + β)q(x)2), (32)where δ(x) = µ(x) and q(x) = v(µ(x))/λ1. So we are led to option (vi)of our theorem.In the switched case (f1(x), g1(x)) = ((αx2 + β)v(x)2, x2), sincedeg f1(x) = 2, v(x) is a constant polynomial andg(x) = P̂n/2(`(g1(µ(x)))) = P̂n/2((`1µ(x)2 + `0)q(x)2), (33)where q(x) ≡ 1. Thus, we arrived again at option (vi) with δ(x) = µ(x)and a = `1, b = `0.Now, if the standard pair (f1(x), g1(x)) is of the third kind overQ, then (f1(x), g1(x)) = (D2(x, αt), Dt(x, α2)) with t being odd. Let us10 ANDRA´S BAZSO´ AND ISTVA´N PINKsubstitute f1(x) = x2−2αt into (28) to deduce that `(x) = (x+2αt)/λ21,whenceg(x) = P̂n/2(`(g1(µ(x)))) = P̂n/2(Dt(µ(x), α2) + 2αtλ21). (34)It follows from Proposition 2 that −2αt/λ21 is an extremum of thepolynomial Dt(µ(x), α2)/λ21, which is of P -type (1, 2, . . . , 2) as t is odd.Hence (Dt(µ(x), α2) + 2αt)/λ21 = δ(x)q(x)2 for some δ(x), q(x) ∈ Q[x]with deg δ(x) = 1. We deduce, that g(x) is of the form (iv).Finally, consider the case when (f1(x), g1(x)) is a standard pair ofthe fourth kind over Q. Then(f1(x), g1(x)) =(D2(x, α)α,Dt(x, β)β(t/2)),with an even t. Substituting this into (28), an easy calculation yields`(x) = (αx+ 2α)/λ21, whence, by (25), we obtaing(x) = P̂n/2(`(g1(µ(x)))) = P̂n/2(αβ−t/2Dt(µ(x), β) + 2αλ21). (35)Now from Proposition 2 we infer that−2βt/2αβ−t/2λ21= −2αλ21is one of the two extrema of the polynomial αβ−t/2Dt(µ(x), β)/(λ21)and it is of P -type (2, 2, . . . , 2) as t is even. Therefore we haveαβ−t/2Dt(µ(x), β) + 2αλ21= q(x)2for some q(x) ∈ Q[x]. Thus g(x) is of the form (iii), which completesthe proof.AcknowledgementsThe first author was supported by the Hungarian Academy of Sci-ences and by the OTKA grant NK104208. The second author waspartially supported by the European Union and the European SocialFund through project Supercomputer, the national virtual lab (grantno: TA´MOP-4.2.2.C-11/1/KONV-2012-0010).DIOPHANTINE EQUATIONS WITH APPELL SEQUENCES 11References1. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions,National Bureau of Standards, 1964.2. P. Appell, Sur une classe de polynoˆmes. Ann. Sci. Ecole Norm. Sup., 9(1880), 119-144.3. A. Bazso´, D. Kreso, F. Luca and A´. Pinte´r, On equal values of powersums of arithmetic progressions, Glas. Mat. Ser. III , 47 (2012), 253–263.4. Y. F. Bilu, Quadratic factors of f(x)−g(y), Acta Arith., 90 (1999), 341–355.5. Y. F. Bilu and R. F. Tichy, The Diophantine equation f(x) = g(y), ActaArith., 95 (2000), 261–288.6. Y. F. Bilu, B. Brindza, P. Kirschenhofer, A´. Pinte´r and R. F. Tichy,Diophantine equations and Bernoulli polynomials (with an Appendix by A.Schinzel), Compositio Math., 131 (2002), 173–188.7. M. Kulkarni and B. Sury, Diophantine equations with Bernoulli polyno-mials, Acta Arith., 116 (2005), 25–34.8. M. Kulkarni and B. Sury, A class of Diophantine equations involving Ber-noulli polynomials, Indag. Math. (N.S.), 16 (2005), 51–65.9. M. Kulkarni and B. Sury, On the Diophantine equation 1 +x+ x22! + . . .+xnn! = g(y)., In: Diophantine equations, 121-134, Tata Inst. Fund. Res. Stud.Math., Mumbai, 2008.10. Cs. Rakaczki, On the Diophantine equation Sm(x) = g(y), Publ. Math.Debrecen, 65 (2004), 439–460.11. Cs. Rakaczki and D. Kreso, Diophantine equations with Euler polynomi-als, Acta Arith., 161 (2013), 267–281.12. S. Roman, The umbral calculus, Academic Press, New York, 1984.Andra´s Bazso´Institute of MathematicsMTA-DE Research Group ”Equations Functions and Curves”Hungarian Academy of Sciences and University of DebrecenP.O. Box 12, H-4010 Debrecen, HungaryE-mail address: bazsoa@science.unideb.huIstva´n PinkInstitute of MathematicsUniversity of DebrecenP.O. Box 12, H-4010 Debrecen, HungaryE-mail address: pinki@science.unideb.hu

Diophantine equations with Appell sequences

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Diophantine equations with Appell sequences

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