Some new continued fraction sequence convergent to the Somos quadratic recurrence constant

Transcription

1 You et al. Journal of Inequalities and Applications (06 06:9 DOI 0.86/s y R E S E A R C H Open Access Some new continued fraction sequence convergent to the Somos quadratic recurrence constant Xu You *, Shouyou Huang and Di-Rong Chen,4 * Correspondence: youxu@bipt.edu.cn Department of Mathematics and Physics, Beijing Institute of Petrochemical Technology, Beijing, 067, P.R. China Full list of author information is available at the end of the article Abstract In this paper, we provide some new continued fraction approximation and inequalities of the Somos quadratic recurrence constant, using its relation with the generalized Euler constant. MSC: 40A05; 40A0; 40A5; 65B0; 65B5 Keywords: Somos quadratic recurrence constant; generalized Euler constant; continued fraction; multiple-correction method Introduction Somos [] defined the sequence g n ngn,withg 0 in 999. Finch []provedtheasymptotic formula in 00 as follows: ( g n σ n n + n + 4 n n + 8 n,09 + (n, 4 n 5 where the constant σ is now nown as the Somos quadratic recurrence constant. This constant appears in important problems by pure representations, σ {... / exp or integral representations, { } { x σ exp 0 ( x ln x dx exp see [ 5]. The generalized-euler-constant function } ln, 0 0 } x ( xy ln(xy dxdy ; γ (z z + ( z ln (. 06 You et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s and the source, provide a lin to the Creative Commons license, and indicate if changes were made.

2 You et al. Journal of Inequalities and Applications (06 06:9 Page of 0 was introduced by Sondow and Hadjicostas [6] and Pilehrood and Pilehrood [7], where γ ( istheclassicalEulerconstant. Sondow and Hadjicostas [6] also defined the generalized Somos quadratic recurrence constant, by ( t /(t σ t exp{ t Since when we set t in(., γ ( ln σ or t(t γ ( }. (. t { σ exp ( } γ, (. these functions are closely related to the Somos quadratic recurrence constant σ.here we denote γ n (z n z + ( z ln. (.4 Recently, many inspiring results of establishing more precise inequalities and more accurate approximations for the Somos quadratic recurrence constant and generalized-eulerconstant function were given. Mortici [8] provided a double inequality of the error estimate by the polynomial approximation. Lu and Song [9]gavesharperbounds. Motivated by this important wor, in this paper we will continue our previous wor [0 ] and apply a multiple-correction method to construct some new sharper double inequality of the error estimate for the Somos quadratic recurrence constant. Moreover, we establish sharp bounds for the corresponding error terms. Notation Throughout the paper, the notation (; x means a polynomial of degree in x with all of its non-zero coefficients positive, which may be different at each occurrence. Estimating γ (/ In order to deduce some estimates for the σ constant, we evaluate the series γ ( ( + ln. (. First we need the following intermediary result. Lemma For every integer positive, we define α (x (+ x x + b 0 c 0 x x+ c x x+ c x+c x, α 4 (x x ( + x + b 0 c 0 x c x+ x c x+ x x+ c x+c x 4, where b 0 8, c , c,689,840, c,759,64 09,5, c,080,065,44,04,8,07, c 4,49,06,5,7,675 06,44,5,688,8. Then for every integer, we have a ( a ( +< ln + < a 4 ( a 4( +. (.

3 You et al. Journal of Inequalities and Applications (06 06:9 Page of 0 Proof Based onourpreviousworwe will applymultiple-correction method and study the double inequality of the error estimate as follows. (Step The initial correction. Because x+ ln x x,wechooseα x (x+ 0(x ( + x b 0 x+c 0. Then letting the coefficient of x 5, x 6 of the molecule in the following fractions equal zero, we have b 0 8, c ,and f 0 (a (x 0 (x ( a x + 0(x + ln x x 6,55,650 7,6,005x 7,8,084x 5,66,x,75,56x 4 x ( + x (69 + 6x (85 + 6x <0. As the molecule in the above fractions has all coefficients negative, we see as a result that f 0 (x is strictly decreasing. (Step The first correction.weletα (x ( + b x 0 x+ c 0 x. Then letting the coefficient of x+c x 6 of the molecule in the following fractions equal zero, we have c,689,840 and As f (a (x (x ( a x + (x + ln x x (5; x x 4 ( + x 4 ( x ( x >0. (5; x 54,756,089,068 +,64,,84,70x +,49,04,987,656x +,48,80,9,88x + 67,774,7,785x 4 +50,67,886,95x 5 has all coefficients positive, we see as a result that f (x is strictly increasing. But f ( 0, so f (x<0on[,. (Step The second correction. Similarly, we let α (x x ( + b 0 x+ c 0 x x+ c x x+c. Then letting the coefficient of x 7 of the molecule in the following fractions equal zero, we have c,759,64 09,5 and f (a (x (x ( a x + (x + ln x x (6; x x ( + x (9, ,0x +,445x (497,47 + 8,00x +,445x <0. As (6; x,996,85,6,7,40,746,76 + 5,859,9,4,468,,987,90x + 4,46,77,746,7,804,785,684x + 4,054,456,699,47,979,755,000x + 6,6,66,07,948,05,05,50x ,4,977,4,95,0,5x 5 +,547,66,888,095,7,400x 6

4 You et al. Journal of Inequalities and Applications (06 06:9 Page 4 of 0 has all coefficients positive, we see as a result that f (x is strictly decreasing. But f ( 0, so f (x>0on[,. (Step 4 The third correction. Similarly, we let α (x b ( + 0 x c 0 x. Then letting x+ x+ c x x+ c x+c x the coefficient of x 8 of the molecule in the following fractions equal zero, we have c,080,065,44,04,8,07 and f (a (x (x ( a x + (x + ln x x (7; x x 4 ( + x 4 (6,055,0 + 5,556,840x +84,89x (,997, ,6,58x +84,89x >0, we see as a result that f (x is strictly increasing. But f ( 0,sof (x<0on[,. This finishes the proof of the left-hand inequality in (.. (Step 5 The fourth correction. Similarly, we let α 4 (x b ( + 0 x c 0 x. Then letting x+ c x+ x c x+ x x+ c x+c x 4 the coefficient of x 9 of the molecule in the following fractions equal zero, we have c 4,49,06,5,7,675 06,44,5,688,8 and f 4 (a (x 4 (x ( a x + 4(x + ln x x (8; x x ( + x (; <0, x (; x we see as a result that f 4 (x is strictly decreasing. But f 4 ( 0,sof 4 (x>0on[,. This finishes the proof of the right-hand inequality in (.. This is the end of Lemma. Remar It is worth to point out that Lemma provides some continued fraction inequalities by the multiple-correction method. Similarly, repeating the above approach step by step, we can get more sharp inequalities. But this maybe brings about some computation increase, the details omitted here. By adding inequalities of the form a ( a ( + < ( + ln < a 4( a 4( + from n +to,weget a (n + n < n+ ( + ln < a 4(n +. (. n These double inequalities give the error estimate when γ is approximated by γ n n + ln. So we have the following theorem.

5 You et al. Journal of Inequalities and Applications (06 06:9 Page 5 of 0 Theorem For every positive integer n, + 8,49, ,5,85n +70,55,590n +,545,70n γ n < γ ( n 0( + n (,997, ,6,58n +84,89n < γ n +,77,456,0,47 +,9,69,079,700n +,447,045,99,090n +,489,45,90n 0( + n (,8,754,5, ,7,674,0n +9,567,5,707n +,749,644,864n. (.4 Proof The double inequality (. can be equivalently written as ( ( a (n + < γ γ n n < a 4(n + n and the conclusion follows if we tae into account that and a (n + 8,49, ,5,85n +70,55,590n +,545,70n n n 0( + n (,997, ,6,58n +84,89n a 4 (n + n,77,456,0,47 +,9,69,079,700n +,447,045,99,090n +,489,45,90n 0( + n (,8,754,5, ,7,674,0n +9,567,5,707n +,749,644,864n. This is the end of Theorem. Remar In fact, the upper and lower bounds in (.4 are sharper than the ones in (. of Mortici [8] and (.8 of Lu and Song [9] for every positive integer n. From (.4 we can provide the following result which has a simpler form than (.4, although it is weaer than (.4. Corollary For every positive integer n, we have γ n + n (n + 8 < γ Proof We tae into account that a (n + (n + 8 ( ( < γ n + n (n +. (.5 5,800,96, ,765,54,466n +,85,5,6n +,87,56,05n +69,7,00n 4 0( + n (8 + n (,997, ,6,58n +84,89n >0

6 You et al. Journal of Inequalities and Applications (06 06:9 Page 6 of 0 for n, and (n + a 4(n + 7,978,40,0,7 + 75,9,80,05,696n + 69,8,,904,n >0 + 4,976,66,6,440n + 749,98,97,800n 4 / ( 0( + n ( + n,8,754,5, ,7,674,0n + 9,567,5,707n +,749,644,864n for n. Combining with Theorem, the conclusion follows. This is the end of Corollary. Combining (. andcorollary, we obtain the following estimates for the Somos quadratic recurrence constant. Corollary For every positive integer n, we have { exp γ n n+ (n + } { < σ <exp γ n n+ (n + 5 }. (.6 Estimating γ (/ Mortici [8] and Lu and Song [9] have provided a double inequality for the error estimate of γ (/. In order to give the new error estimate for γ (/, we need the following intermediary result. Lemma For every integer positive, we define b ( ( + x b 4 ( ( + x x + x + d 0 0 x x+ x x+ x x+, d 0 0 x x+ x x+ x x+ x x+ 4, where d 0 9 4, 0 8 5, 9 405,,880,55 444,58, 905,967,495 4,8,848, 4,856,400,40,4 7,57,975,660,55. Then, for every integer, we have b 4 ( b 4( +< ln + < b ( b ( +. (. Proof Based on our previous wor we will apply multiple-correction method to study the double inequality of the error estimate as follows. (Step The initial correction. Because x+ ln x x,wechooseb x x (x+ 0(x ( + d x 0 x+ 0. Then letting the coefficient of x 6, x 7 of the molecule in the following frac-

7 You et al. Journal of Inequalities and Applications (06 06:9 Page 7 of 0 tions equal zero, we have d 0 9 4, 0 8 5,and g 0 (b (x 0 (x b x + 0(x + ln x x x 85,64 +,57,90x +,84,69x +,,8x +69,865x 4 +9,00x 5 4x 4 ( + x 4 (8 + 5x ( + 5x >0, we see as a result that g 0 (x is strictly increasing. (Step The first correction. Weletb (x ( + d x 0. Then letting the coefficient x+ 0 x x+ of x 7 of the molecule in the following fractions equal to zero, we have and g (b (x (x b x + (x + ln x x x (6; x 4x 5 ( + x 5 (70 + 8x (45 + 8x. As (6; x has all coefficients positive, we see as a result that g (x is strictly decreasing. (Step The second correction. Similarly, we let b (x d (+ 0. Then we let the x x+ 0 x x+ x x+ coefficient of x 8 of the molecule in the following fractions equal zero, we have,880,55 444,58 and g (b (x (x b x + (x + ln x x x (7; x 4x 4 ( + x 4 (75,0 + 44,565x +49,9x (,6,67 + 5,49x +49,9x >0, we see as a result that g (x is strictly increasing. (Step 4 The third correction.weletb (x ( + x d 0 x+ 0 x x+ x x+ x x+. Then letting the coefficient of x 9 of the molecule in the following fractions equal zero, we have 905,967,495 4,8,848 and g (b (x (x b x + (x + ln x x x (8; x/ ( 4x 5 ( + x 5( 6,595,045 +,576,565x +,05,094x ( 9,4,704 +,68,75x +,05,094x. As (8; x has all coefficients positive, we see as a result that g (x is strictly decreasing. But g ( 0,sog (x>0on[,. This finishes the proof of the right-hand inequality in (..

8 You et al. Journal of Inequalities and Applications (06 06:9 Page 8 of 0 (Step 5 The fourth correction. Similarly, we let b 4 (x x ( + d 0 x+ 0 x x+ x x+ x x+ x x+ 4. Then letting the coefficient of x 0 of the molecule in the following fractions equal zero, we have 4,856,400,40,4 7,57,975,660,55 and g 4 (b (x 4 (x b x + 4(x + ln x x x (9; x 4x 4 ( + x 4 (; >0, x (; x we see as a result that g 4 (x is strictly increasing. But g 4 ( 0,sog 4 (x<0on[,. This finishes the proof of the left-hand inequality in (.. This is the end of Lemma. By adding inequalities of the form b 4 ( b 4( + < ( + ln ( < b ( b ( + from n +to,weget b 4 (n + n < n+ ( + ln Combining equations (. and(.4, we have γ ( ( γ n n+ n+ ( < b (n +. (. n ( + ln. (. Using inequality (. and equality (., we have the following theorem. Theorem For every positive integer n, where γ n < γ < γ n + (; n n 40( + n (; n + n+ 86,097, ,06,48n +47,7,70n +,06,880n n+ n 40( + n 4 (9,4,704 +,68,75n +,05,094n, (.4 (; n 0,676,544,766,567 +,86,057,64,00n +,74,044,504,0n + 69,,9,400n,

9 You et al. Journal of Inequalities and Applications (06 06:9 Page 9 of 0 (; n,6,90,65,79 + 8,07,79,047n + 56,7,54,606n + 8,455,696,60n. Proof If we tae into account that b (n+ n and b 4(n+ n, combining equations (.and(., the conclusion follows. This is the end of Theorem. From (.4 we can provide another result, which has a simpler form than (.4, although it is weaer than (.4. Corollary For every positive integer n, we have γ n Proof We use the bounds b 4 (n ++ for n, where n 4(n + + n (n + (n +4 < γ (n + (; x 40(( + n (; x >0 (. (.5 (; x 9,77,08,45,07 +,96,877,797,870n +,50,057,00,00n + 69,,9,400n, (; x,6,90,65,79 + 8,07,79,047n + 56,7,54,606n +8,455,696,60n, and the telescoping inequalities n (n + n n ((n + +n + < n n < n (n + n n ((n + +n +. Combining Theorem, the conclusionfollows. This is the end of Corollary. Remar It is worth to point out that the multiple-correction method provides a general way to find some continued fraction approximation of σ t for t >. Similarly, repeating the above approach step by step, we can get more sharp inequalities. But this maybe brings about some computation increase, the details omitted here. Competing interests The authors declared that they have no competing interests. Authors contributions The authors read and approved the final manuscript.

10 You et al. Journal of Inequalities and Applications (06 06:9 Page 0 of 0 Author details Department of Mathematics and Physics, Beijing Institute of Petrochemical Technology, Beijing, 067, P.R. China. School of Mathematics and Statistics, Hubei Normal University, Huangshi, Hubei 4500, P.R. China. Department of Mathematics, Wuhan Textile University, Wuhan, Hubei 4000, P.R. China. 4 School of Mathematics and System Science, Beihang University, Beijing, 009, P.R. China. Acnowledgements This wor was supported by the National Natural Science Foundation of China (Grant Nos and Computations made in this paper were performed using Mathematica 9.0. Received: 8 January 06 Accepted: March 06 References. Somos, M: Several constants related to quadratic recurrences. Unpublished note (999. Finch, SR: Mathematical Constants. Cambridge University Press, Cambridge (00. Guillera, J, Sondow, J: Double integrals and infinite products for some classical constants via analytic continuations of Lerch s transcendent. Ramanujan J. 6(,47-70 ( Ramanujan, S: Collected Papers of Srinivasa Ramanujan. Edited by Hardy, GH, Aiyar, PVS, Wilson, BM. Am. Math. Soc., Providence ( Sloane, NJA: Sequences A059, A0, A44, and A660 in the On-Line Encyclopedia of Integer Sequences 6. Sondow, J, Hadjicostas, P: The generalized Euler-constant function γ (z and a generalization of Somos quadratic recurrence constant. J. Math. Anal. Appl. (, 9-4 ( Pilehrood, KH, Pilehrood, TH: Arithmetical properties of some series with logarithmic coefficients. Math. Z. 55(, 7- ( Mortici, C: Estimating the Somos quadratic recurrence constant. J. Number Theory 0, (00 9. Lu, D, Song, Z: Some new continued fraction estimates of the Somos quadratic recurrence constant. J. Number Theory 55, 6-45 (05 0. Cao, XD, Xu, HM, You, X: Multiple-correction and faster approximation. J. Number Theory 49, 7-50 (05. Cao, XD: Multiple-correction and continued fraction approximation. J. Math. Anal. Appl. 44, (05. You, X: Some new quicer convergences to Glaisher-Kinelin s and Bendersy-Adamchi s constants. Appl. Math. Comput. 7, -0 (05. You, X, Chen, D-R: Improved continued fraction sequence convergent to the Somos quadratic recurrence constant. J. Math. Anal. Appl. 46, 5-50 (06