Reflection Symmetries of q-bernoulli Polynomials

Transcription

1 Journal of Nonlinear Matematical Pysics Volume 1, Supplement 1 005, 41 4 Birtday Issue Reflection Symmetries of q-bernoulli Polynomials Boris A KUPERSHMIDT Te University of Tennessee Space Institute Tullaoma, TN 37388, USA bkupers@utsi.edu Tis article is part of te special issue publised in onour of Francesco Calogero on te occasion of is 70t birtday Abstract A large part of te teory of classical Bernoulli polynomials B n x s follows from teir reflection symmetry around x = 1/: B n 1 x = 1 n B n x. Tis symmetry not only survives quantization but as two equivalent forms, classical and quantum, depending upon weter one reflects around 1/ te classical x or quantum [x] q. 1 Introduction Te classical Bernoulli polynomials are te difference analogs of te indefinite integrals of te power functions. Tese polynomials are defined by te generating function Since Bx;t = B n x tn n! = t e t 1 ext. 1.1 B1 x; t = Bx;t : 1. B1 x; t = t e t 1 e1 x t = we find tat = t e t 1 ext = Bx;t, te t e t 1 e t ext = B n 1 x = 1 n B n x, n Z Te question is: wat appens wit te reflection symmetry 1.3 in te q-domain, wen one considers q-bernoulli polynomials? Originally q-bernoulli numbers and polynomials were introduced by Carlitz in 1948 [1], but tey do not seem to be te most natural ones; in particular, tey don t appear as te values at nonpositive integers of various Riemann and Hurwitz q-zeta functions see [4]. Te particular version of q-bernoulli polynomials wic is used in te next four Sections, up to a rescaling, can be sown to be equivalent to te one defined by Tsumura from different considerations and by different formulae [5,6]. Copyrigt c 005 by B A Kupersmidt

2 Reflection Symmetries of q-bernoulli Polynomials 413 q-notations We sall be working wit te following quantization sceme: if x is a classical object, suc as a complex number, its q-version [x] q is defined as [x] q = qx 1 = ex 1 e 1,.1 were is considered as a fixed complex number or a formal parameter: q = e, q 0, Te q-numbers [x] q satisfy many simple relations, all easily verified, suc as: [x + y] q = [x] q + q x [y] q,.4 [x] q 1 = q 1 x [x] q,.5 [ x] q = q x [x] q,.6 [ x] q 1 = q[x] q,.7 [1 x] q = 1 [x] q q-bernoulli polynomials Let ˆβt be te generating function of te classical Bernoulli numbers: ˆβt = B n n! tn = t e t We sall write ˆβ for te formal power series ˆβ = B n n. 3. n! Tis formal differential operator acts properly on polynomials in x were it terminates ˆβ x k = and also on exponents: B n n x k = n! k k B n x k n = B k x, 3.3 n ˆβ e tx = ˆβte tx = Bx;t. 3.4

3 414 B A Kupersmidt Now set: B n x q = ˆβ Bx;t q = [x] n, n Z 0, 3.5 B n x q tn n! = ˆβ e t[x], 3.6 were [x] stands for [x] q were tere is no risk of confusion. Tese are our q-versions of te classical Bernoulli polynomials. Since x [x] q = x + + O, 0, 3.7 we see tat B n x q = B n x + n B n+1x B n x + O, 3.8 wen 0. Te first few q-bernoulli polynomials are: B 0 x q = 1, B 1 x q = B x q = q 1 [x] q + ˆβ 1 [x] q [x] q + q te corresponding q-bernoulli numbers, tus, are: 3.9, 3.10 ; 3.11 B 0 0 q = 1, 3.1 B 1 0 q = q + 1, 3.13 In general, so tat B 0 q = q q [x] q n = n q x 1 n = n n B n x q = n n j=0 From formula 3.5 we find: B n x + 1 q B n x q = [x] q n = j=0 n q jx 1 n j, 3.15 j n 1 n j ˆβjq jx j qx n[x] q n

4 Reflection Symmetries of q-bernoulli Polynomials 415 Tis is our particular version of te classical difference equation B n x + 1 B n x = nx n Anoter classical relation, db n x dx = nb n 1 x, 3.19 is easily seen to be quantized into te form: B n x q = nb n 1x q + nb n x q. 3.0 All tese formulae can be profitably viewed from te point of view of te umbral calculus, as te properties of te objects U e t[x], 3.1 were Ut = u n t n n! 3. is an arbitrary formal power series. We won t need suc an interpretation for arguments tat follow, and it is left as an Exercise to te reader. 4 Te expected form of te reflection symmetry Let s consider te effect of reflection at x = 1/ on te q-bernoulli polynomials. We ave: B1 x;t q = ˆβ = ˆβ = ˆβ expt[1 x] q [by 4.3] = expt[1 x] q = ˆβ exp exp tq 1 [x] q 1 = Bx; tq 1 q 1 : expt[ x] q [by.7] = B1 x;t q = Bx; tq 1 q In components, tis is: B n 1 x q = q n B n x q 1, n Z 0, 4. our first q-analog of te classical reflection formula 1.3. In te Proof above, we used te classical formula

5 416 B A Kupersmidt ˆβ t = e t ˆβt. 4.3 It as te following q-analog. Let ˆβt q = B0; t q 4.4 be te generating function of te q-bernoulli numbers. Ten ˆβ t q = e t/q ˆβq t q In oter words, te combination e t/ ˆβqt q 4.6 is invariant wit respect to te involution t t, q q Formulae 4.1, 4.5, and 3.17 imply tat e t ˆβqt q ˆβt q = ˆβt, 4.8 wic is wat becomes of te classical definition ˆβt = t/e t 1 under quantization. We won t need below te last 5 formulae. 5 Te unexpected form of te reflection symmetry To make te argument more transparent, we first separate te variable x from te quantization parameter q. Set X = [x] q, 5.1 and define te polynomials B ext n X q by te rule: B n x q = B ext n [x] q q. 5. For example, formulae yield: B ext 0 X q = 1, B ext 1 X q = B ext X q = q 1 X + ˆβ 1 5.3, 5.4 X X + q q

6 Reflection Symmetries of q-bernoulli Polynomials 417 Tus, wile B n x q is a polynomial of degree = n in q x, Bn ext X q is a polynomial of degree = n in X; te coefficients in bot cases are functions of and q wic are polynomial in and rational in q. We are going now to consider reflection at 1/ of X, not x. If we suggestively denote by te result of reflection at X = 1/ on functions of X, wit te simultaneous cange q q 1 : X = 1 X, q = q 1, 5.6 fx q = f1 X q 1, 5.7 and if we remember tat X = [x] q = 1 q x 1, 5.8 te X = 1/ reflection 5.6 becomes, in te x-language: X = [x ] q 1 = q x 1 q 1 1 = q1 q x = 1 X = 1 [x] q = 1 qx 1 = q qx = q q1 x = x = 1 x. 5.9 Tus, B ext n 1 X q = q n B ext n X q 1, n Z 0, 5.10 Te nature of te nd reflection formula 5.10 is somewat mysterious. I ave stumbled on it by an accident, trying to understand a still bigger mystery: proper quantum analogs of te Faulaber penomenon see [3,]. 6 Perspectives Te Bernoulli polynomials form just one subclass among te large class of objects connected to various zeta functions and Eisenstein series. It is natural to inquire if te reflection symmetry survives quantization of tis class. It is indeed so for te case of 1-dimensional circular Eisenstein series see [7]. Even more important is te question of wat appens wit te reflection symmetry of Bernoulli polynomials under quantization scemes different from te one given by formula.1, suc as te aritmetic quantizations [x] = qx q rx q q r, r Z 0, 6.1

7 418 B A Kupersmidt or even te general two-parameter quantization family [x] q,q = qx Q x q Q, q Q 6. and its quasiclassical limit [x] quas q = lim Q q q x Q x q Q = xqx An analysis of te derivation of formula 4.1 sows tat te reflection symmetry x 1 x survives a given quantization sceme {x [x] q } provided tere exists an involution I : q q 6.4 in te space of quantization parameters q = q 1,...,q N suc tat te ratio [ x]q/[x]q = ρq 6.5 is x-independent. In suc a case, te q-bernoulli polynomials possess te symmetry were B n 1 x q = ρq n B n x q, 6.6 B n x q = ˆβ [x]q n, n Z For te two-parameter case 6., so tat [ x] q,q /[x] q 1,Q 1 = q 1 Q 1, 6.8 B n 1 x q,q = qq n B n x q 1,Q 1, 6.9 and we recover formula 4. for Q = 1. Tere is no quantum form of te reflection symmetry for te variable X = [x] q,q ; tat variable sould be considered as not te rigt one. Te correct coice is: X = 1 + ρq 1/ [x 1 ] q Indeed, applying te involution : x 1 x,q q to formula 6.10, we find: X = 1 + ρq 1/ [ 1 x] q But formula 6.5 tells us tat

8 Reflection Symmetries of q-bernoulli Polynomials 419 ρq = ρq 1, 6.1 and terefore, ρq 1/ [ y]q + ρq 1/ [y]q = ρq 1/ ρq[y]q + +ρq 1/ [y]q = 0, y = 1 x Tus, X + X = 1, 6.14 and we recover te quantum form of te reflection symmetry. For te simplest quantum sceme we ave started wit, [x] q = q x 1/, te general recipe 6.10 produces a different variable X new, tan te old variable X old = [x] q : X new X old = 1 q1/qx 1/ + 1 qx 1 = 1 q1/ 1 = = 1 [1 ] q; 6.15 and te equality sows tat [ 1 ] q + [ 1 ] q 1 = q1/ 1 + q 1/ 1 q 1 1 = q1/ 1 qq 1/ 1 = X new X old + X new X old = I conclude wit a remark on oter quantization scemes, a fascinating and largely unexplored subject. One suc sceme, a sort of second quantization of te [x] q - sceme, is: 1[x] ;Q = 1 + x Q 1, 6.18 were n n q = 1 + Q j, n Z 1 1, n = 0 and, in general, x Q = 1 + Q /1 +Q x Q, 0 < Q < 1, 6.0a 1 + x Q 1 = 1 +Q 1 x x Q, 0 < Q < b

9 40 B A Kupersmidt For Q = 1, formula 6.18 becomes: 1[x] ;1 = 1 + x 1 = qx 1 = [x] q, q = 1 +, 6.1 te familiar formula.1. However, te quantization sceme 6.18 is easily seen to lack te reflection symmetry x 1 x, since te formula 6.5 is not satisfied. Neverteless, since [ ] x 1 + x Q = 1 + [x] Q + Q + O 3, 0, 6. we ave: [ ] x 1[x] ;Q = [x] Q + Q and we can set: Q < x > ;Q = [x] Q + Q [ x Q + O, 0, 6.3 ], = 0, 6.4 Q a sort of quasiclassical extension of te quantum sceme {x [x] Q }. Miraculously, te quantization sceme 6.4 reacquires te reflection symmetry {x x = 1 x} under te involution because Q = Q 1, =, 6.5 ρ;q = < x > ;Q < x > ;Q = [ x] Q1 + Q [ x 1] Q /[] Q [x] Q 11 Q [x 1] Q 1/[] Q 1 = = Q Q 1 + Q Q x Q Q x+1 1 Q Q 1 Q 1 1 = Q A few comments are in order. First, if we set q = 1 +, = 0, 6.7 in te quantization sceme 6., we find: [x] ;Q = [x] Q x, = 0, 6.8 Q 1 [x] Q wic is quite different from formula 6.4. Second, one can restore te reflection symmetry to te 1 + x Q - quantization sceme by taking [x] ;Q ;Q as eiter 1 + x Q x Q 6.9

10 Reflection Symmetries of q-bernoulli Polynomials 41 or 1 + x Q 1 + x Q, 6.30 and coosing te involution ;Q ;Q at will. Tird, at Q = 1, formula 6.4 becomes: x < x > = x +, = Tis is te simplest possible deformation, a nilpotent one, of te usual number system, and it suggests a very particular way to calculate te 1-jet of classical matematics, a sort of an even version of te supersymmetric extension. Fourt, if we apply te perturbation form 6.4 to te quantization sceme [x] q,q = q x Q x /q Q : [ ] x 1 < x > ;q,q = [x] q;q +, = 0, 6.3 q,q te reflection symmetry criterion 6.5 no longer olds. It does reappear wen we cange formula 6.3 into < x > ;q,q = [x] q,q 1 + [] q,q α[x] q,q, = 0, 6.33 wit α being an arbitrary constant, and te involution in te quantum parameter space {;q,q} being given by te formulae q = q 1, Q = Q 1, = /qq α Tese subjects will be dealt wit on anoter occasion. Acknowledgement I am very grateful to A. P. Veselov for carefully reading te preliminary versions of tis paper and suggesting very useful corrections. References [1] Carlitz, L., q-bernoulli Numbers And Polynomials, Duke Mat, J [] Fairlie, D. B., and Veselov, A. P., Faulaber and Bernoulli Polynomials and Solitons, Pys. D [3] Knut, D.E., Joann Faulaber and Sums of Powers, Mat. Comp [4] Kupersmidt, B.A., You Are A Bad Zeta Function, Aren t You? to appear

11 4 B A Kupersmidt [5] Tsumura, H., A Note On q-analogues Of Te Diriclet Series And q-bernoulli Numbers, J. Number Teory [6] Tsumura, H., A Note On q-analogues Of Diriclet Series, Proc. Japan Acad. 75 Ser. A [7] Weil, A., Elliptic Functions According To Eisenstein and Kronecker, Springer-Verlag New York, 1976.