2311Article 07.1.7Journal of Integer Sequences, Vol. 10 (2007),236147Deformations of the Taylor FormulaEmmanuel FerrandInstitut Math´ematique de JussieuUMR 7586 du CNRSUniversit´e Pierre et Marie Curie4, place Jussieu75252 Paris [email protected] a sequence x = {xn, n ∈ N} with integer values, or more generally withvalues in a ring of polynomials with integer coefficients, one can form the generalizedbinomial coefficients associated with x,¡nm¢x=Qml=1xn−l+1xl. In this note we introduceseveral sequences that possess the following remarkable feature: the fractions¡nm¢xarein fact polynomials with integer coefficients.1 IntroductionBy a deformation of the integers we mean a sequence x = {xn, n ∈ N} of polynomials in oneor more variables and with integral coefficients, having the property that there exists somevalue q0of the variables such that ∀n ∈ N, xn(q0) = n. The quantum integers xn=Pn−1l=0qlare a typical example of a deformation of the integers. Another example is given by theversion of the Chebyshev polynomials defined by xn(cos(θ)) =sin(nθ)sin(θ).In this note we consider some deformations of the factorial function and of the binomialcoefficients that are induced by such deformations of the integers. This situation can beinterpreted as a deformation of the Taylor formula, as explained below. Given a polynomialP of degree n with complex coefficients, the Taylor expansion at some point X givesP (X + 1) = P (X) + 1 ·dPdX(X) +122!·d2PdX2(X) + · · · +1nn!·dnPdXn(X).In other words, if one denotes by τ : C[X] → C[X] the “translation by one” operator,defined by τ (P )(X) = P (X + 1), then τ = exp(ddX). A matrix version of this fact can1be stated as follows. Denote by P and D the semi-infinite matrices whose coefficients are,respectively, Pi,j=¡ij¢and Di,j= i if i = j + 1 and 0 otherwise, (i, j) ∈ N2. ThenP = exp(D).P =1 0 0 0 . . .1 1 0 0 . . .1 2 1 0 . . .1 3 3 1 . . ................D =0 0 0 0 . . .1 0 0 0 . . .0 2 0 0 . . .0 0 3 0 . . ................This suggests the following way to deform the Taylor formula. Replace the sequence Nof the integers which appears as the non-zero coefficients of D by the terms of a sequencex = {xn, n ∈ N} with values in some polynomial ring. Denote by Dxthe correspondingmatrix. Given some integer n, define n!xto be the polynomial n!x=Qnl=1xl. Define expxtobe the formal series expx(t) =P∞k=0tkk!x. Observe that the matrix expx(Dx) is well definedsince, coefficients-wise, the summation is finite. Its coefficients expx(Dx)i,jwill be denotedby the symbols¡ij¢xand will be called the generalized binomial coefficients associated withthe sequence x. Note thatµij¶x=jYl=1xi−l+1xlif i ≥ j, and 0 otherwise.This definition has appeared already in several contexts; see, for example, Knuth andWilf [4] for an introduction to the relevant literature. Note that the fractions¡ij¢xhave no apriori reason to be polynomials with integer coefficients. In fact, such a phenomenon appearsonly for very specific sequences x.In this note we are interested in deformations of the integers x that possess this property.The first part of the paper (section2) is a variation on the classical theme of quantumintegers and q-binomials. It deals with sequences that satisfy a second order linear recurrencerelation. In the second part, (section3), we deform the integers and the q-binomials in a lessstandard way, using a sequence that satisfies a first order non-linear recurrence relation. Inthe third part (section4), we introduce a sequence related to the Fermat numbers (which isnot a deformation of the integers), and we show that the corresponding generalized binomialcoefficients are polynomials with integer coefficients.Let us mention that Knuth and Wilf [4] showed that if a sequence x with integral valuesis a gcd-morphism (that is, xgcd(n,m)= gcd(xn, xm)), then the associated binomial coefficientsare integers.2 q-binomialsThe properties of the so-called “quantum integers”[n]q=n−1Xl=0ql=1 − qn1 − q2and the associated “q-binomials” were investigated long before the introduction of quantummechanics (see [2]). We rephrase below an approach developed by Carmichael [1] (andprobably already implicit in earlier works). It deals with a slightly more general, two-variableversion of the quantum integers.Consider the sequence x with values in Z[a, b] defined by the following linear recurrencerelation of order 2:x0= 0, x1= 1, xn+1= a · xn+ b · xn−1.This sequence specializes to the quantum integers when a = q + 1 and b = −q (and tothe usual integers for a = 2 and b = −1).Remark. xnis given by the following explicit formula:xn=nXl=1µl − 1n − l¶a2l−n−1bn−las one can check by induction. 2Proposition 1. (rephrased from [1]).• x : N → Z[a, b] is a gcd-morphism:gcd(xn, xm) = xgcd(n,m).• The associated binomial coefficients¡nm¢xare polynomials in a and b with integral co-efficients.2The first few rows of the corresponding deformation of Pascal’s triangle are as follows:1 0 0 0 0 01 1 0 0 0 01 a 1 0 0 01 a2+ b a2+ b 1 0 01 a3+ 2ba (a2+ 2b)(a2+ b) a3+ 2ba 1 01 a4+ 3ba2+ b2(a4+ 3ba2+ b2)(a2+ 2b) (a4+ 3ba2+ b2)(a2+ 2b) a4+ 3ba2+ b21Many classical sequences of integers or polynomials arise as solutions of second orderrecurrence relations with the appropriate initial conditions. The corresponding deformationsof the Pascal triangle have often been considered separately in the literature. They receivea unified treatment through Carmichael’s approach.3Example 1. For a = b = 1, the sequence x specializes to the Fibonacci sequence(A000045 in [5]), and the triangle looks as follows:1 0 0 0 0 0 . . .1 1 0 0 0 0 . . .1 1 1 0 0 0 . . .1 2 2 1 0 0 . . .1 3 6 3 1 0 . . .1 5 15 15 5 1 . . .......................Example 2. For a = 3 and b = −2, the sequence x specializes to the Mersenne numbers(A000225 of [5]), xn= 2n− 1. The triangle then looks like1 0 0 0 0 0 . . .1 1 0 0 0 0 . . .1 3 1 0 0 0 . . .1 7 7 1 0 0 . . .1 15 35 15 1 0 . . .1 31 155 155 31 1 . . .......................Example 3. For a = 2s and b = −1, the sequence xn= Un−1(s), where Unis then−th Chebyshev polynomial of the second kind. This implies that, for any (n, m)