On an iteration leading to a q-analogue of the Digamma function

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1 On an iteration leading to a q-analogue of the Digamma function Christian Berg and Helle Bjerg Petersen May, 22 Abstract We show that the q-digamma function ψ q for < q < appears in an iteration studied by Berg and Durán. This is connected with the determination of the probability measure ν q on the unit interval with moments / n+ ( q/( qk, which are q-analogues of the reciprocals of the harmonic numbers. The Mellin transform of the measure ν q can be expressed in terms of the q-digamma function. It is shown that ν q has a continuous density on ],], which is piecewise C with kinks at the powers of q. Furthermore, ( qe x ν q (e x is a standard p-function from the theory of regenerative phenomena. 2 Mathematics Subject Classification: primary 33D5; secondary 44A6. Keywords: q-digamma function, Hausdorff moment sequence, p-function. Introduction For a measure µ on the unit interval [, ] we consider its Bernstein transform B(µ(z = as well as its Mellin transform M(µ(z = t z dµ(t, Rz >, ( t t z dµ(t, Rz >. (2 These functions are clearly holomorphic in the right half-plane Rz >. Corresponding author

2 The two integral transformations are combined in the following theorem from [4] about Hausdorff moment sequences, i.e., sequences (a n n of the form a n = for a positive measure µ on the unit interval. t n dµ(t, (3 Theorem. Let (a n n be a Hausdorff moment sequence as in (3 with µ. Then the sequence (T(a n n defined by T(a n n = /(a a n is again a Hausdorff moment sequence, and its associated measure T(µ has the properties T(µ({} = and B(µ(z + M( T(µ(z = for Rz >. (4 This means that the measure T(µ is determined as the inverse Mellin transform of the function /B(µ(z +. It follows by Theorem. that T maps the set of normalized Hausdorff moment sequences (i.e., a = into itself. By Tychonoff s extension of Brouwer s fixed point theorem, T has a fixed point (m n. Furthermore, it is clear that a fixed point (m n is uniquely determined by the equations Therefore giving ( + m m n m n =, n. (5 m 2 n+ + m n+ m n =, (6 m = , m 2 =, Similarly, T maps the set M + ([, ] of probability measures on [, ] into itself. It has a uniquely determined fixed point ω and m n = t n dω(t, n =,,.... (7 Berg and Durán studied this fixed point in [5],[6], and it was proved that the Bernstein transform f = B(ω is meromorphic in the whole complex plane and characterized by a functional equation and a log-convexity property in analogy with Bohr-Mollerup s characterization of the Gamma function, cf. [2]. Let us also mention that ω has an increasing and convex density with respect to Lebesgue measure m on the unit interval. An important step in the proof of these results is to establish that ω is an attractive fixed point so that in particular the iterates T n (δ converge weakly 2

3 to ω. Here and in the following δ a denotes the Dirac measure with mass concentrated in a R. It is easy to see that T(δ = m, because T(,,... n = n + = t n dt. It is well-known that the Bernstein transform of Lebesgue measure m on [, ] is related to the Digamma function ψ, i.e., the logarithmic derivative of the Gamma function, since t z t dt = ψ(z + + γ = z, Rz >, (8 n(n + z n= cf. [, 8.36]. Here γ = ψ( is Euler s constant. Therefore ν := T(m = T 2 (δ is determined by M(ν (z = B(m(z + = ψ(z γ. The measure ν = T(m is given explicitly in [4] as ν = ( α n t ξn dt, (9 n= where ξ =, ξ n (n, n +, n =, 2,... is the solution to ψ( ξ n = γ and α n = /ψ ( ξ n. The moments of the measure ν are the reciprocals of the harmonic numbers, i.e., ( t n dν (t = n+ =. ( H n+ k The purpose of the present paper is to study the first two elements of the sequence T n (δ q, where < q < is fixed. The reason for excluding q = is that T(δ = δ. Since ω is an attractive fixed point, we know that the sequence converges weakly to ω. The first step in the iteration is easy: T(δ q = ( q q k δ q k, ( k= because t z d T(δ q (t = q q = ( q q k q kz. (2 z+ k= 3

4 This shows that T(δ q is the Jackson d q t-measure on [, ] used in the theory of q-integrals, cf. [9]. It is a q-analogue of Lebesgue measure in the sense that d q t m weakly for q. It is therefore to be expected that ν q := T(d q t = T 2 (δ q is a q-analogue of the measure ν, and we are going to determine ν q. We have M(ν q (z = f q (z +, (3 where f q is defined as the Bernstein transform of d q t: ( t z f q (z = t d qt = ( q z + q k qkz. (4 q k This formula is a q-analogue of (8 and the relationship between f q and q-versions of Euler s constant and the Digamma function is given in (2. The moments of ν q are q-analogues of ( ( n t n q dν q (t =. (5 q k+ k= It is easily seen that q tn dν q (t is an increasing function mapping [, ] onto [(n +, Hn+ ]. Formally ν q is the inverse Mellin transform of /f q (z +, i.e., ν q = 2πi i i t z f q (z + dz, and we shall exploit that in section 3, where we transfer the harmonic analysis on the multiplicative group ], [ to the additive group of real numbers, hence replacing the Mellin transformation by the ordinary Fourier transformation. If we denote by τ q the image measure of ν q under the transformation log(/t, we formally get τ q (x = e iyx dy 2π f q ( + iy, (6 but since y /f q ( + iy is a square integrable and non-integrable function as shown in Section 3, this only yields that τ q is a square integrable function. In Theorem 3. we prove that τ q is the restriction to [, [ of a continuous symmetric positive definite function. The main tool to obtain further regularity properties of ν q will be a direct approach using convolution. In fact, we realize ( qτ q as the convolution of an exponential density and an elementary kernel N n, see (26, which makes it possible to prove that τ q (x = e (+cqx p n (x, x [n log(/q, (n + log(/q], n =,,... 4

5 for a certain constant c q, cf. (22, and a polynomial p n of degree n the coefficients of which are given explicitly as functions of q, see (3. In Remark 2.2 we point out that ( qτ q (x is a standard p-function in the sense of []. Going back to the interval ], ] we can state our main result that ν q is a C -spline, i.e. it has a piecewise C -density with respect to Lebesgue measure on [, ]: Theorem.2 The measure ν q has a continuous and piecewice C -density denoted ν q (t on ], ]. We have ν q (t = t cq p n (log(/t, t [q n+, q n ], n =,,..., (7 where c q is given by (22 and p n is a polynomial of degree n given by (3. The derivative of ν q has a jump of size /( q n ( q at the point q n, n =, 2,.... Remark.3 It follows that the behaviour of ν q (t is oscillatory, and therefore quite different from that of ν (t given by (9, which is increasing and convex. See Figure and 2 which shows the graph of ( qν q for q =.5 and q =.9. Figure : The graph of ( qν q (t on [q 3, ] for q =.5 2 Proofs Jackson s q-analogue of the Gamma function is defined as Γ q (z = (q; q (q z ; q ( q z, 5

6 Figure 2: The graph of ( qν q (t on [q 3, ] for q =.9 cf. [9], and its logarithmic derivative ψ q (z = d dz log Γ q(z = log( q + log q k= q k+z q k+z (8 has been proposed in [2] as a q-analogue of the Digamma function ψ. See also the recent paper [3]. We define the q-analogue of Euler s constant as γ q = ψ q ( = log( q log q The Bernstein transform f q of d q t is given in (4, hence q k qk. (9 f q (z q = z + = z + = z + q k q kn ( q kz n= ( (q k(n+ q k(n++z n= log(/q (γ q + ψ q (z +, showing that f q (z = ( qz + q log(/q (γ q + ψ q (z +, (2 6

7 so f q has a close relationship with the q-digamma function and the q-version of Euler s constant. We will be using another expression for f q (z/( q derived from (4, namely with f q (z q = z + c q c q = q k q kqkz, (2 q k qk. (22 Clearly, q/( q < c q < q/( q 2 for < q < and q c q is a strictly increasing map of ], [ onto ], [. We mention two other expressions c q = d(nq n = n= ( (q n ; q, where d(n is the number of divisors in n, see [8, p. 4]. In order to replace the Mellin transformation by the Laplace transformation we introduce the probability measure τ q on [, [ which has ν q as image measure under t e t, hence L(τ q (z = n= e tz dτ q (t = f q (z +. The analogue of Theorem.2 about the measure τ q is given in the next theorem, which we shall prove first. Theorem 2. The measure τ q has a continuous density also denoted τ q with respect to Lebesgue measure on [, [. It is C in each of the open intervals ]n log(/q, (n + log(/q[, n =,,... with jump of the derivative of size J n = q 2n ( q n ( q (23 at the point n log(/q, n =, 2,.... Furthermore, τ q ( = /( q and lim t τ q (t =. Proof of Theorem 2.. Introducing the discrete measure µ = q 2k q kδ k log(/q of finite total mass µ = c q q/( q < c q, (24 7

8 we can write hence q f q (z + = f q (z + q = + c q + z L(µ(z, ( ( + c q + z( L(µ(z + c q + z = n= (L(µ(z n ( + c q + z n+. (25 Let ρ q denote the following exponential density restricted to the positive half-line ρ q (t = exp( ( + c q ty (t, where Y is the usual Heaviside function equal to for t and equal to zero for t <. Its Laplace transform is given as e tz ρ q (t dt = ( + c q + z, but this shows that (25 is equivalent to the following convolution equation ( qτ q = ρ q (µ ρ q n = n= n= ρ (n+ q µ n. (26 This equation expresses a factorization of ( qτ q as the convolution of the exponential density ρ q and an elementary kernel N n with N = µ ρ q. For information about the basic notion of elementary kernels in potential theory, see [7, p.]. All three measures in question τ q, ρ q and (µ ρ q n are potential kernels on R in the sense of [7]. The measure µ n, n is a discrete measure concentrated in the points k log(/q, k = n, n +,.... The convolution powers of ρ q are Gamma densities ρ (n+ q (t = tn n! e (+cqt Y (t, as is easily seen by Laplace transformation. Clearly, ρ q µ is a bounded integrable function with ρ q µ ρ q µ < c q, ρ q µ = ρ q µ < c q + c q, (27 and then ρ q (ρ q µ n, n is a continuous integrable function on R, vanishing for t n log(/q and for t. Furthermore, ρ q (ρ q µ n < (c q /( + c q n, and this shows that the right-hand side of (26 converges uniformly on [, [, so ( qτ q has a continuous density on [, [ tending to at infinity. Note that ( qτ q ( = ρ q ( =. 8

9 For n and x [n log(/q, [ we get ρ (n+ q x µ n (x (x t n = e (+cq(x t Y (x t dµ n (t n! = e (+cqx (x k log(/q n q k(+cq Y (x k log(/qµ n (k log(/q n! k=n which is a finite sum, and µ n (k log(/q = p +...+p n=k j= n q 2p j, k = n, n +,.... q p j In particular, ( q µ n 2 n (n log(/q =, µ n ((n + log(/q = q For n and x < (n + log(/q we then get nq 2n+2 ( q n ( + q. ( qτ q (x = (28 n j e (+cqx q j(+cq (x j log(/q k Y (x j log(/q µ k (j log(/q. k! j= On [, log(/q] it is equal to exp( ( + c q x, on [log(/q, 2 log(/q] it is equal to ( exp( ( + c q x + q cq (x log(/q, q on [2 log(/q, 3 log(/q[ it is equal to ( exp( ( + c q x + q cq (x log(/q+ q q 2( cq q2( cq (x 2 log(/q + q2 2( q 2(x 2 log(/q2. k= On [n log(/q, (n + log(/q], n we can write ( qτ q (x (29 n j = e ( (+cqx + q j(+cq (x j log(/q k µ k (j log(/q, k! j= because µ (j log(/q = for j. 9

10 This shows that τ q (x = e (+cqx p n (x, x [n log(/q, (n + log(/q], n =,,..., (3 where p n is the polynomial of degree n given by ( p n (x = n j + q j(+cq (x j log(/q k µ k (j log(/q. (3 q k! j= The derivative of the expression (29 is ( + c q ( qτ q (x (32 n j + e (+cqx q j(+cq (x j log(/q k µ k (j log(/q, (k! j= and the value R n at the point x = n log(/q, n is R n ( n j = q [ ( n(+cq + c q + q j(+cq ((n j log(/q k µ k (j log(/q k! j= ] n j + q j(+cq ((n j log(/q k µ k (j log(/q (k! j= ( n j = q [ ( n(+cq + c q + q j(+cq ((n j log(/q k µ k (j log(/q k! + n q j(+cq j= j= j ((n j log(/q k µ k (j log(/q (k! + q n(+cq µ(n log(/q ]. The value L n+ of (32 at x = (n + log(/q is L n+ = q (n+(+cq [ ( n j ( + c q + q j(+cq ((n + j log(/q k µ k (j log(/q k! j= ] n j + q j(+cq ((n + j log(/q k µ k (j log(/q. (k! j= The difference R n L n = q 2n /( q n

11 is the jump of ( qτ q at x = n log(/q, and this gives the jump J n of (23. To transfer the results of Theorem 2. to give Theorem.2, we use that ν q is the image measure of τ q under t e t, hence We then get ν q (t = (/tτ q (log(/t, < t. and similarly hence D + ν q (t t=q n = t 2 [D τ q (log(/t + τ q (log(/t] t=q n = q 2n [D τ q (n log(/q + τ q (n log(/q], D ν q (t t=q n = q 2n [D +τ q (n log(/q + τ q (n log(/q], [D + ν q (t D ν q (t] t=q n = q 2n (D +τ q (n log(/q D τ q (n log(/q, and the assertion follows. Remark 2.2 The representation (25 and Theorem 2. show that ( qτ q is a standard p-function in the terminology from the theory of regenerative phenomena. This follows from [, Theorem 3.], and the measure µ + (/( qδ plays the role of the measure µ in []. The size J n of the jump of the derivative at n log(/q equals µ({n log(/q} = This is in accordance with [, Theorem 3.4]. q2n q n. Figure 3 and 4 show the graph of ( qτ q for q =.5 and q =.9. 3 Further properties of τ q Formally, by Fourier inversion we get that τ q (x = 2π e iyx dy f q ( + iy.

12 Figure 3: The graph of ( qτ q (x on [, 3 log(/q] for q =.5 The function /f q ( + iy is a non-integrable L 2 -function, so the formula holds in the L 2 -sense. To see this we notice that where In particular f q (z z = q + h q (t = ( q f q ( + iy + iy = q + e tz h q (t dt, Rz >, k>t/ log(/q q k qk. (33 e ity e t h q (t dt, and since e t h q (t is integrable, it follows from the Riemann-Lebesgue Lemma that we get the asymptotic behaviour Furthermore, from (2 we get f q ( + iy ( q( + iy, y. (34 hence Rf q ( + iy = q + ( q Rf q ( + iy q + q k q k ( q k cos(ky log(q, q k ( + q k, 2

13 Figure 4: The graph of ( qτ q (x on [, 3 log(/q] for q =.9 showing that Rf q ( + iy is bounded below and above. It follows that the symmetrized density τ q (x if x, ϕ q (x = (35 τ q ( x if x <, is the Fourier transform of the non-negative integrable function 2Rf q ( + iy f q ( + iy 2, and therefore ϕ q (x is continuous and positive definite, so τ q is the restriction to [, [ of such a function. Summing up we have proved Theorem 3. For x τ q (x = 4 cos(xy Rf q( + iy f q ( + iy 2 dy is the restriction of a continuous symmetric positive definite function (35. For < t tν q (t = 4 cos(y log t Rf q( + iy f q ( + iy dy. 2 3

14 Remark 3.2 The function f q defined in (4 is a Bernstein function in the sense of [7], but not a complete Bernstein function in the sense of [4], because f q (z/z is not a Stieltjes function as shown by formula (33. This is in contrast to lim f q(z = ψ(z + + γ, q which is a complete Bernstein function, cf. [4]. 4 Relation to other work The transformation T can be extended from normalized Hausdorff moment sequences to the set K = [, ] N of sequences (x n = (x n n of numbers from the unit interval [, ]. This was done in [3], where T : K K is defined by (T(x n n = + x x n, n. (36 The connection is that a normalized Hausdorff moment sequence (a n n is considered as the element (a n n K. Since T is a continuous transformation of the compact convex set K in the space R N of real sequences equipped with the product topology, it has a fixed point by Tychonoff s theorem, and this is (m n n. There is no reason a priori that the fixed point (m n of (36 should be a Hausdorff moment sequence, but as we have seen above, the motivation for the study of T comes from the theory of Hausdorff moment sequences. Although T is not a contraction on K in the natural metric d((a n, (b n = 2 n a n b n, (a n, (b n K, n= it was proved in [3] that T maps K into the compact convex subset C = { (a n K a 2}, and the restriction of T to C is a contraction. It is therefore possible to infer that (m n is an attractive fixed point from the fixed point theorem of Banach. Acknowledgment The first author wants to thank Fethi Bouzzefour, Tunesia for having raised the question of determining the measure ν q. References [] Akhiezer, N. I., The classical moment problem. Oliver and Boyd, Edinburgh,

15 [2] Artin, E., The Gamma Function. Holt, Rinehart and Winston, New York 964. [3] Berg, C., Beygmohammadi, M., On a fixed point in the metric space of normalized Hausdorff moment sequences, Rend. Circ. Mat. Palermo, Serie II, Suppl. 82 (2, [4] Berg, C., Durán, A. J., Some transformations of Hausdorff moment sequences and Harmonic numbers, Canad. J. Math. 57 (25, [5] Berg, C., Durán, A. J., The fixed point for a transformation of Hausdorff moment sequences and iteration of a rational function, Math. Scand. 3 (28, 39. [6] Berg, C., Durán, A. J., Iteration of the rational function z /z and a Hausdorff moment sequence, Expo. Math. 26 (28, [7] Berg, C., Forst, G., Potential theory on locally compact abelian groups. Ergebnisse der Mathematik und ihrer Grenzgebiete Band 87. Springer- Verlag, Berlin-Heidelberg-New York, 975. [8] Fine, N. J., Basic hypergeometric series and applications. American Mathematical Society, Providence, RI, 988. [9] Gasper, G., Rahman, M, Basic hypergeometric series. Cambridge University Press, Cambridge 99, second edition 24. [] Gradshteyn, I. S., Ryzhik, I. M., Table of integrals, series and products, Sixth Edition, Academic Press, New York, 2. [] Kingman, J. F. C., Regenerative Phenomena. John Wiley & Sons Ltd., London 972. [2] Krattenthaler, C., Srivastava, H. M., Summations for Basic Hypergeometric Series Involving a q-analogue of the Digamma Function, Computers Math. Applic. 32 (996, [3] Mansour, T., Shabani, A. S., Some inequalities for the q-digamma function, J. Inequal. Pure and Appl. Math. (29, Art. 2, 8 pp. [4] Schilling, R., Song, R. and Vondraček, Z., Bernstein functions: Theory and Applications, de Gruyter, Berlin 2. Christian Berg, Helle Bjerg Petersen Department of Mathematical Sciences University of Copenhagen Universitetsparken 5 5

16 DK-2 København Ø, Denmark addresses: (C. Berg (H. B. Petersen. 6