Little q-legendre polynomials and irrationality of certain Lambert series

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1 Little -Legendre olynomials and irrationality of certain Lambert series arxiv:math/0087v [math.ca 23 Jan 200 Walter Van Assche Katholiee Universiteit Leuven and Georgia Institute of Technology June 8, 208 Abstract Certain -analogs h () of the harmonic series, with = / an integer greater than one, were shown to be irrational by Erdős [9. In Peter Borwein [4 [5 used Padé aroximation and comlex analysis to rove the irrationality of these -harmonic series and of -analogs ln (2) of the natural logarithm of 2. Recently Amdeberhan and Zeilberger [ used the EKHAD symbolic acage to find -WZ airs that rovide a roof of irrationality similar to Aéry s roof of irrationality of ζ(2) and ζ(3). They also obtain an uerboundfor the measureof irrationality, but better uer bounds were earlier given by Bundschuh and Väänänen[8 and recently also by Matala-aho and Väänänen [4 (for ln (2)). In this aer we show how one can obtain rational aroximants for h () and ln (2) (and many other similar uantities) by Padé aroximation using little -Legendre olynomials and we show that roerties of these orthogonal olynomials indeed rove the irrationality, with an uer boundof the measure of irrationality which is as shar as the uer bound given by Bundschuh and Väänänen for h () and a better uer bound as the one given by Matala-aho and Väänänen for ln (2). Introduction Most imortant secial functions, in articular hyergeometric functions, have -extensions, usually obtained by relacing Pochhammer symbols (a) n = a(a+) (a+n ) by their -analog (a;) n = ( a)( a)( a 2 ) ( a n ). Since lim ( a ;) n ( ) n = (a) n, one usually retrieves the original secial function from its -extension by letting. A good source for -extensions of hyergeometric series (basic hyergeometric series) is the boo [0 by Gaser and Rahman. Our interest in this aer is the following -extension of the harmonic series h () = = =, 0 < = WVA is a Research Director of the Belgian Fund for Scientific Research (FWO). = <, ()

2 and of the natural logarithm of 2 ln (2) = = ( ) = = ( ), 0 < = <. (2) In 948, Paul Erdős [9 roved that h 2 () is irrational. Peter Borwein [4 [5 showed that h () (and other similar numbers) are irrational for every integer > and also roved the irrationality of ln (2) for every integer >. If we denote E (z) = (z;), with = /, then Bézivin [3 had earlier shown that E (α),e (α),...,e() (α) are linearly indeendent over Q for every N and α Q with α j for every integer j. The case = corresonds to irrationality of α+ j. j= Borwein used Padé aroximation techniues and comlex analysis to obtain good rational aroximants to h () and ln (2). Indeed, one can use the following lemma to rove irrationality [6, Lemma 5.: Lemma Let x be a real number, and suose there exist integers a n,b n (n N) such that. x a n /b n for every n N, 2. lim (b n x a n ) = 0, then x will be irrational. This lemma exresses the fact that the order of aroximation of a rational number by rational numbers is one and not higher [2, Theorem 86. Furthermore, if x a n /b n = O(/b +s n ) with 0 < s < and b n < b n+ < bn +o(), then the measure of irrationality r(x) (Liouville-Roth number, order of aroximation) r(x) = inf{r R : x a/b < /b r has at most finitely many integer solutions (a,b)} satisfies 2 r(x) + /s (see, e.g., [6, exercise 3 on. 376 for the uer bound; the lower bound follows since every irrational number is aroximable to order 2 [2, Theorem 87). Recently Amdeberhan and Zeilberger [ found -WZ airs to obtain rational aroximants for h () and ln (2), imroving the uer bound for the measure of irrationality to 4.8 = 24/5. However, four years earlier Bundschuh and Väänänen [8,. 78hadestablished better uerbounds: r(h ()) +(π 2 +2)/(π 2 2) = and r(ln (2)) +(2π 2 +3)/(π 2 3) = The uer bound for r(ln (2)) was imroved by Matala-aho and Väänänen [4 to Inthisaerweshowthatonecanfindrationalaroximantswhicharerelatedtolittle -Legendre olynomials and hence return to Padé aroximation. We can then use some results for little -Legendre olynomials to rove the irrationality once more, and with the aid of some elementary number theory we obtain the same bound for the measure of irrationalityastheoneobtainedbybundschuh andväänänenforh ()andabetteruer bound as the one given by Matala-aho and Väänänen for ln (2). The connection with little -Legendre olynomials oens the way for roving the irrationality of -extensions of ζ(2) and ζ(3) in Aéry s sirit [2, using multile orthogonal -olynomials [6. 2

3 2 Little -Legendre olynomials The little -Legendre olynomials are defined by ( ) P n (x ) = 2 φ n, n+ ;x ( n ;) ( n+ ;) x =, (;) (;) 0 < <, (3) and they are orthogonal olynomials on the exonential lattice {, = 0,,2,...}: P m ( )P n ( ) = n 2n+δ n,m. (4) If we use the -binomial coefficients [ n = (;) n (;) (;) n, (5) and the formulas ( n ;) = ( ) n+( )/2 (;) n (;) n, ( n+ ;) = (;) n+ (;) n, (6) then (3) reduces to and since P n (x ) = [ [ n n+ lim [ n = n+(+)/2 ( x), (7) ( ) n, we see that we indeed find the Legendre olynomials on [0, by letting tend to limp n (x ) = ( n )( n+ ) ( x) = P n (x). Let = / so that > whenever 0 < <, then [ n (;) = ( ) (+)/2 (;), = (n )[ n, (8) and we can rewrite the little -Legendre olynomial as P n (x ) = [ [ n n+ n+( )/2 ( x). (9) 3

4 There is a Rodrigues formula for these little -Legendre olynomials in terms of the - difference oerator D for which namely f(z) f(z) if z 0, D f(z) = ( )z f (0) if z = 0, (0) P n (x ) = n(n )/2 ( ) n (;) n D n [(x;) nx n, () which will be useful later. We refer to [3 for more information and references for little -Legendre olynomials. Euation (9) exresses the little -Legendre olynomials in the basis {,x,x 2,...,x n } of monomials. Sometimes it is more convenient to use another basis of olynomials, and for orthogonal olynomials on {, = 0,,2,...} a convenient set of basis functions is {(x;), = 0,,2,...,n}. We will need to use some -series for this urose. Recall the -analog of Newton s binomium formula (x;) n = [ n ( )/2 ( x), (2) and its dual x n = [ n ( ) n+(+)/2 (x;). (3) A more general result is the -binomial series n=0 (a;) n (;) n x n = (ax;) (x;), <, x <. (4) Use (3) with argument n+ x in the Rodrigues formula (), then we have One finds easily that P n (x ) = n(n+) ( ) n (;) n D (x;) n = so that, using (8) we find the reuired exansion P n (x ) = ( ) n [ n ( ) n+(+)/2 D (x;) n n+. (;) n (;) n ( ) (x;) n, (5) [ [ n n+ ( ) (n )(n +)/2 (x;). (6) 4

5 3 The -harmonic series Orthogonal olynomials naturally arise in Padé aroximation of a Stieltjes function dµ(x) f(z) =, z / su(µ). (7) z x Suose µ is a ositive measure on the real line with infinite suort and for which all the moments exist. If P n (x) (n = 0,,2,...) are orthogonal olynomials for µ, i.e., P n is of degree n and P n (x)p m (x)dµ(x) = 0, m n, and if Q n are the olynomials of degree n given by Pn (z) P n (x) Q n (z) = dµ(x), (8) z x then it is easy to see that P n (z)f(z) Q n (z) = If we exand /(z x) around z = as then by orthogonality we have Pn (x) dµ(x), z / su(µ). (9) z x n z x = x xn + z+ z n z x, P n (z)f(z) Q n (z) = z n Pn (x)x n z x dµ(x) = O(/zn+ ). This is recisely the (linearized) form of the interolation conditions near z = for Padé aroximation so that Q n (z)/p n (z) is the [ n Padé aroximant for f(z) near z =. n For little -Legendre olynomials the measure µ is suorted on {, = 0,,2,...}, which is a bounded set in [0, with one accumulation oint at 0. The measure is given by g(x)dµ(x) = g( ). The Stieltjes function for this measure is f(z) = We will need this function at n, where it gives z = z. (20) f( n ) = n+ = h n (). (2) = 5

6 Hence if > is an integer, then f( n ) gives h () u to n = /( ), which is a rational number. Now use (9) for little -Legendre olynomials at z = n to find P n ( n ) ( h () n = ) Q n ( n ) = P n ( ) n. (22) Observe that (9) gives P n ( n ) = [ [ n n+ n ( ) ( )/2, (23) whichisnearlytheb n foundin[,.277(theirb n corresondstop n ( n+ )). Observealso [ that Borwein s construction [5, Lemma 2 uses P n (c n+ ). The -binomial numbers n are olynomials in with integer coefficients, which follows easily from the -version of Pascal s triangle identities [ [ [ [ [ n n n n n = + = + n, hence if > is an integer then [ n and [ n+ are integers. This means that (23) imlies P n ( n ) to be an integer. Furthermore, since n > and all the zeros of P n (x ) are in [0,, we also may conclude that ( ) n P n ( n ) is ositive for all n. The second imortant uantity in (22) is Q n ( n ). This associated little -Legendre olynomial can be comuted exlicitly using (8) and is given by Q n (x ) = j=0 P n (x ) P n ( j ) x j j. Use (6) to write this as Q n (x ) = ( ) n Now use [ [ n n+ ( ) (n )(n +)/2 j=0 (x;) ( j+ ;) x j j. (x;) (y;) x y = l (y;) l ( l+ x;) l, l= which one can easily rove by induction, then this gives [ [ n n+ Q n (x ) = ( ) n+ ( ) (n )(n +)/2 l ( l+ x;) l j ( j+ ;) l. l= j=0 6

7 Using the -binomial series (4) we can calculate the modified moments so that j ( j+ ;) l = (;) l j(l ;) j (;) j j=0 Q n (x ) = ( ) n+ Evaluating at x = n, and using then gives Q n ( n ) = ( ) n+ [ [ n n+ j=0 = (;) l ( l+ ;) (;) = l, ( ) (n )(n +)/2 l= ( l+ n ;) l = ( n ;) l, [ [ n n+ ( ) (n )(n +)/2 l= ( l+ x;) l. (24) l ( n ;) l. (25) l All the terms in the sum for Q n ( n ) are now integers, excet for the l in the denominators. In order to obtain an integer we therefore need to multily everything by a multile of all l for l =,2,...,n. We will choose d n () = n Φ (), (26) = where Φ n (x) = n = gcd(,n)= (x ω n ), ω n = e 2πi/n, (27) are the cyclotomic olynomials [5, 4.8. Each cyclotomic olynomial is monic, has integer coefficients, and the degree of Φ n is φ(n) (Euler s totient function). It is nown that x n = d n Φ d (x), (28) and that every cyclotomic olynomial is irreducible over Q[x. Hence d n () is a multile of all l for l =,2,...,n. The growth of this seuence is given by the following lemma, which was essentially given by A. O. Gel fond, who obtained the uer bound in [, Euation (7). We give a roof to mae this aer self-contained. 7

8 Lemma 2 Suose is an integer greater than one and let d n () be given by (26). Then lim d n() /n2 = 3/π2. (29) Proof: The degree of d n () as a monic olynomial in is n =ϕ() and an old result of Mertens (874) shows that for n this grows lie [2, Theorem 330 ϕ() = 3 +O(nlogn). (30) π 2n2 = We will adat the classical roof of (30) to rove our lemma. For this we use Möbius inversion of (28) to find the reresentation Φ n (x) = d n(x d ) µ(n/d), where µ is the Möbius function. Taing logarithms in (26) gives logd n () = µ(/d)log( d ). = Changing the order of summation gives Now so that m d= logd n () = l= d log( d ). n/l µ(l) d= log( d ) = log m(m+)/2 (;) m = m(m+) 2 logd n () = n2 2 log l= ( µ(l) +O n l 2 = n2 log 2ζ(2) +O(nlogn). The lemma now follows by using ζ(2) = π 2 /6. So far we have found that the numbers log+log(;) m, l= ) l are integers and b n = d n ()P n ( n ), (3) n a n = d n ()Q n ( n )+b n. (32) b n h () a n = d n () = P n ( ) n. (33) 8

9 We now want to show that b n h () a n 0 for all n and lim (b nh () a n ) = 0, so that Lemma imlies the irrationality of h (). First observe that P n ( ) n = P n ( n ) If we add and subtract P n ( ) in the sum, then we have P n ( ) n = P n ( n ) P n ( )P n ( n ) n. P n ( ) P n( n ) P n ( ) + n P n ( n ) The first sum on the right hand side vanishes because of the orthogonality, so that b n h () a n = d n() P n ( n ) P 2 n ( ) n. P 2 n ( ) n. (34) All the terms in the sum are now ositive, and ( ) n P n ( n ) is ositive for all n, hence we may conclude that ( ) n (b n h () a n ) > 0, n =,2,... Next we show that this uantity converges to zero. Clearly n n n so that n Pn 2 ( ) P 2 n ( ) n n Pn 2 ( ). Now we can use the norm of the little -Legendre olynomial (4) to find 2n+ P 2 n ( ) n n+ ( n )( 2n+ ). (35) What remains is to find the asymtotic behavior of P n ( n ) as n. For this we can use a very general theorem for seuences of olynomials with uniformly bounded zeros. Lemma 3 Suose P n (n N) is a seuence of monic olynomials of degree n and that the zeros x j,n ( j n) of P n are such that x j,n M, with M indeendent of n. Then we have for x > and every c C Proof: Factoring the olynomial P n gives lim P n(cx n ) /n2 = x. (36) P n (x) = n x x j,n. j= 9

10 We have the obvious bounds hence when cx n > M For n large enough this easily gives which gives the desired result. x M x x j,n x +M, ( cx n M) n P n (cx n ) ( cx n +M) n. ( x c M ) /n P x n n (cx n ) /n2 x ( c + M ) /n, x n Observe that we may allow M to grow with n subexonentially. For little -Legendre olynomials the zeros are all in [0, so that we can use the Lemma with M =. The leading coefficient κ n of P n (x ) is, by (9), eual to κ n = ( ) n[ 2n n n(n+)/2, giving lim κ n /n2 =, and hence Lemma 3 gives for x > and c C lim P n(cx n ) /n2 = x. (37) Theorem Suose > is an integer. Let a n and b n be given by (3) (32), then a n Z, ( ) n b n N, and ( ) n (b n h () a n ) > 0 for n >. Furthermore lim b nh () a n /n2 = 3(π 2 2) 2π 2 <, which imlies that h () is irrational with measure of irrationality r 2π2 π 2 2 = Proof: If we tae x = and c = in (37) then we have lim P n( n ) /n2 = 3/2, and combining with (29) we have for the integer b n in (3) lim b n /n2 = 3(π 2 +2) 2π 2. Observe that b n has the same sign as P n ( n ) which is ( ) n. Furthermore (34) and (35) show that lim b nh () a n /n2 = lim d n () /n2 = 3(π 2 2) lim P n ( n 2π ) 2 <. /n2 0

11 The irrationality now follows from Lemma. Observe that this gives rational aroximants a n /b n for h () satisfying h () a n = O b n ( 3(π 2 2) 2π 2 +ǫ)n 2 for every ǫ > 0. Now b n = 3n2 (π 2 +2)/(2π 2 )+o(n 2), hence h () a n = O b n b n b +π2 2 π 2 +2 ǫ n forevery ǫ > 0, which gives forthemeasure of irrationalitytheboundr + π2 +2 π 2 2 = 2π2 π 2 2. The uer bound for the measure of irrationality is better than the uer bound 4.8 obtained in [, but the same as the (earlier) uer bound of Bundschuh and Väänänen [8. 4 The -analog of the logarithm of 2 Next we show that a very similar analysis also roves the irrationality of ln (2) for every integer >. First of all we rewrite ln (2) using the geometric series ln (2) = ( ) = ( ) j. = Fubini s theorem allows us to change the order of the sums whenever 0 < < and this gives ln (2) = ( ) (j+) = = j=0 = j=0 = = j+ + j+ +. Hence if we evaluate the Stieltjes function (20) at z = n then we find f( n ) = j=0 n+ + = ln n (2)+ +, so thatf( n ) gives thereuired ln (2)uto n = /( +), which isarationalnumber. We can now roceed as in the revious section and evaluate (9) for little -Legendre olynomials at z = n to find P n ( n ) ( ln (2)+ n = = ) Q n ( n ) = + P n ( ) n +.

12 Here we can use (9) to see that P n ( n ) = [ [ n n+ ( )/2, is a ositive integer, and if we use (24) and ( n l+ ;) l = ( n ;) l then Q n ( n ) = ( ) n+ [ [ n n+ ( ) (n )(n +)/2 l= ( n ;) l. l This uantity has the numbers l (l =,2,...,n) in the denominator, so that we need to multily it by a multile of all l with l n. Now we have an additional uantity n = + and in order to mae this an integer we need to multily it by a multile of all + for =,2,...,n. If we choose ˆd n () = n Φ ( 2 ), = then because of 2 = ( )( +) we see that ˆd n () is a multile of all + and all for =,...,n. Note that ˆd n () = d n ( 2 ), where d n is given by (26), so that Lemma 2 gives the growth So if we choose lim d n( 2 ) /n2 = 6/π2. (38) then a n and b n are integers and b n = d n ( 2 )P n ( n ), (39) n a n = d n ( 2 )Q n ( n ) b n +, (40) b n ln (2) a n = d n ( 2 ) = P n ( ) n j. (4) Theorem 2 Suose > is an integer. Let a n and b n be given by (39) (40), then a n Z, b n N, and b n ln (2) a n < 0. Furthermore lim b nln (2) a n /n2 = 3(π 2 4) 2π 2 <, which imlies that ln (2) is irrational. Its measure of irrationality satisfies r 2π2 π 2 4 =

13 Proof: Use c = and x = in (37), then together with (38) we find Furthermore, we have lim b/n2 P n ( ) n j = n = 3(π 2 +4) P n ( n ) 2π 2 n 2. P 2 n ( ) n j and n n + j n + for every j, combined with (4) imlies d n ( 2 ) n+ P n ( n ) ( n +)( 2n+ ) (b nln (2) a n ) d n( 2 ) P n ( n ) 2n+. From this one easily finds the reuired asymtotics and the irrationality then follows from Lemma. Observe that b n = 3n2 (π 2 +4)/(2π 2 )+o(n 2) and ln (2) a n = O b n b +π2 4 π 2 +4 ǫ n forevery ǫ > 0, which gives forthemeasure of irrationalitytheboundr + π2 +4 π 2 4 = 2π2 π 2 4. The uer bound is better than the uer bounds 4.8(obtained in[), 4.3(obtained in [8), and (obtained in [4). 5 Extensions The construction of rational aroximants for h () and ln (2) can be extended with little effort to series of the form L = c, = where c = a/b is a rational number and c for every. Indeed, these series can be obtained by evaluating the Stieltjes function f in (20) at c n, giving We then get where P n (c n ) ( f(c n ) = L n = P n (c n ) = = n c c. = ) Q c n (c n ) = [ [ n n+ ( )/2 ( c), P n ( ) c n, 3

14 and Q n (c n ) = ( ) n+ [ [ n n+ ( ) (n )(n +)/2 l= (c n ;) l. l In order to have integers, the uantities P n (c n ) and Q(c n ) now need to be multilied by b n and by a multile of all l for l n and of all a b for n. A ossible factor is b 2n d n ()(c;) n. This factor grows lie lim b2n d n ()(c;) n /n2 = 3 π In a way similar to the roof of Theorems and 2 we can then rove: Theorem 3 Suose > is an integer and c = a/b is rational but c for every =,2,... Let a n and b n be given by Then a n,b n Z, and b n = b 2n d n ()(c;) n P n (c n ), (42) n a n = b 2n d n ()(c;) n Q n (c n )+b n c. (43) = lim b nl a n /n2 = π 2 3 π 2 <, which imlies that the infinite sum L is irrational. Its measure of irrationality satisfies r 3π2 π 2 3 = The uer bound for the measure of irrationality corresonds to the uer bound given by Bundschuh and Väänänen [8,. 78. For the cases c = and c =, which we handled in Theorems and 2, one can find better uer bounds. Note that the results in [8 and [4 are also valid for and c in other number fields. Our main urose in this aer, however, was to emhasise the use of little -Legendre olynomials in the construction of rational aroximants for certain imortant Lambert series. Acnowledgments This research was carried out while visiting Georgia Institute of Technology. The author wishes to than the School of Mathematics for its hositality. Also thans to a referee for ointing out references [3, [8, [, and [4, and to Roberto Costas for a useful conversation that led to Lemma 2. This research is artially funded by FWO research roject G and INTAS References [ T. Amdeberhan, D. Zeilberger, -Aéry irrationality roofs by -WZ airs, Adv. Al. Math. 20 (998),

15 [2 R. Aéry, Irrationalité de ζ(2) et ζ(3), Astérisue 6 (979), 3. [3 J-P. Bézivin, Indéendance linéaire des valeurs des solutions transcendantes de certaines éuations fonctionelles, Manuscrita Math. 6 (988), [4 P. Borwein, On the irrationality of, J. Number Theory 37 (99), n +r [5 P. Borwein, On the irrationality of certain series, Proc. Cambridge Philos. Soc. 2 (992), [6 J.M.Borwein, P. B.Borwein, Pi and the AGM A Study in Analytic Number Theory and Comutational Comlexity, Wiley, New Yor, 987. [7 P. Borwein, T. Erdélyi, Polynomials and olynomial ineualities, Graduate Texts in Mathematics 6, Sringer-Verlag, New Yor, 995. [8 P. Bundschuh, K. Väänänen, Arithmetical investigations of a certain infinite roduct, Comositio Math. 9 (994), [9 P. Erdős, On arithmetical roerties of Lambert series, J. Indiana Math. Soc. 2 (948), [0 G. Gaser, M. Rahman, Basic Hyergeometric Series, Encycloedia of Mathematics and its Alications 35, Cambridge University Press, Cambridge, 990. [ A. O. Gel fond, Functions which tae on integral values, Mat. Zam. No. 5 (967), ; translated in Math. Notes (967), [2 G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, 938 (Fifth Edition, 979). [3 R. Koeoe, R. F. Swarttouw, The Asey-scheme of hyergeometric orthogonal olynomials and its -analogue, Faculty of Technical Mathematics and Informatics Reort 98-7, Technical University Delft, 998. Available at ft://ft.twi.tudelft.nl/twi/ublications/tech-reorts/998/dut-twi-98-7.s.gz [4 T. Matala-aho, K. Väänänen, On aroximation measures of -logarithms, Bull. Australian Math. Soc. 58 (998), 5 3. [5 J. Stillwell, Elements of Algebra, Undergraduate Texts in Mathematics, Sringer, Berlin, 994. [6 W. Van Assche, Multile orthogonal olynomials, irrationality and transcendence, Contemorary Mathematics [7 A. van der Poorten, A roof that Euler missed..., Aéry s roof of the irrationality of ζ(3), Math. Intell. (979), Deartment of Mathematics Katholiee Universiteit Leuven Celestijnenlaan 200 B B-300 Leuven BELGIUM walter@wis.uleuven.ac.be 5