Closed form expressions for two harmonic continued fractions

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1 University of Wollongong Research Online Faculty of Engineering and Information Sciences - Papers: Part B Faculty of Engineering and Information Sciences 07 Closed form expressions for two harmonic continued fractions Martin W. Bunder University of Wollongong, mbunder@uow.edu.au Joseph Tonien University of Wollongong, dong@uow.edu.au Publication Details Bunder, M. W. & Tonien, J. 07). Closed form expressions for two harmonic continued fractions. The Mathematical Gazzette, 0 55), Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library: research-pubs@uow.edu.au

2 Closed form expressions for two harmonic continued fractions Keywords expressions, fractions, two, continued, closed, form, harmonic Disciplines Engineering Science and Technology Studies Publication Details Bunder, M. W. & Tonien, J. 07). Closed form expressions for two harmonic continued fractions. The Mathematical Gazzette, 0 55), This journal article is available at Research Online:

3 Closed form expressions for two harmonic continued fractions Martin Bunder, Joseph Tonien Introduction. A continued fraction is an expression of the form a 0 + a + b 0 b a + b The expression can continue for ever, in which case it is called an infinite continued fraction, or it can stop after some term, when we call it a finite continued fraction. For irrational numbers, a continued fraction expansion often reveals beautiful number patterns which remain obscured in their decimal expansion. The interested reader is referred to [] for a collection of many interesting continued fractions for famous mathematical constants. When we truncate a continued fraction after some number of terms, we get what is called a convergent. In particular, the following finite continued fraction a 0 + a + b 0 b + b n is called the n th convergent of the above mentioned continued fraction. If the limit, as n approaches, of the n th convergent exists, we say that the infinite continued fraction converges and the limit is its value. If in the above continued fraction all numerators b i are, we denote it by [a 0, a, a, a 3,... ], and its n th convergent by [a 0, a, a,..., ]. If in addition the coefficients are positive integers, it is not hard to prove that the infinite continued fraction always converges. Continued fractions of this type have applications in cryptography [, 3]. When the coefficients are positive real numbers, there is a classical theorem due to Seidel and Stern [4, 5], dating back to the 840s: The Seidel-Stern Theorem. [6, 7] If > 0 then [a 0, a, a, a 3,... ] converges if, and only if, diverges. Since the harmonic series n diverges, by the Seidel-Stern Theorem, the infinite continued fraction [ t, t, t 3,... ] converges for any positive real number t. Let us call these continued fractions the harmonic continued fractions and denote them by HCF t) t +. t + t 3 + When t and t, the values of the harmonic continued fractions are known.

4 Theorem HCF ) + + π Theorem HCF ) + + ln The two harmonic continued fractions in Theorem and Theorem are derived from the following continued fractions of Euler [8]: π, ln. Euler s proof involved the Wallis product and differential equations. In this paper, we offer an elementary and direct proof of Theorem and Theorem. Our proof uses the Euler-Wallis recurrence formula to establish a closed form formula for the convergents of the continued fractions HCF ) and HCF ). Theorem 3 For any natural number n, n n n n n), 3... n ) 4... n) n )

5 Theorem 4 For any natural number n, Euler-Wallis recurrence formulas n n i0 ) i. i + )i + ) The following theorem due to William Brouncker [60-684], the first President of the Royal Society is called the fundamental theorem of continued fractions. It gives us recursive formulas to calculate the numerator p n and the denominator of the convergents. John Wallis [66-703] and Leonhard Euler [ ] in [8] made extensive use of these formulas which are now called the Euler-Wallis formulas. Theorem 5 For any n 0, the n th convergent can be determined as [a 0, a, a,..., ] p n where the sequences {p n } n and { } n are specified as follows p 0, p, p n p n + p n, n 0, q, q 0, +, n 0. The theorem can be proved easily by induction, since [a 0, a, a,...,, + ] [a 0, a, a,..., + ] + ) + + p n + p n ) p n + p n ) + p n + + ) + +p n + p n + + p n+ +. Sometimes we want to investigate the sub-sequences {p n }, {p n }, { }, { } and the following theorem is useful in those scenarios. Theorem 6 The convergents [a 0, a,..., ] pn p n satisfy the following: ) ) p n p n 4, n, 4, n. 3

6 Proof. By the Euler-Wallis formula, The relation for is proved similarly. Finding HCF) p n p n 3 + p n 4, n, p n 3 p n p n 4, p n p n + p n 3 p n + p n p n 4, p n p n + p n p n + p n p n + a ) n + p n p n 4 + p n, p n 4. The following theorem establishes closed form formulas for the numerator p n and the denominator of the convergents of HCF ). Our Theorem 3 follows from this theorem. Theorem 7 The numerator p n and the denominator of the convergents of HCF ) are p n p n+ n i n+ i i +, n 0, i i +, n 0, i 4... n + ) ) n n n + ) i 4... n + ) ) n+ + n n + ) i i +, n 0, i i +, n 0. i Proof. By Theorem 6, p n nn + ) + n ) n + + p n n n + p n 4 n + nn + ) p n n n + p n 4, n So p n 8n + nn + ) p n n n + p n 4, n, p n+ 8n + 8n + 3 n + )n + ) p n n n + p n 3, n. For each n, let p n p n n i i + i and p n+ p n+ n+ i i +, i 4

7 then This simplifies to and so, n + n n n p n 8n + nn + ) n + 3 n + n + n p n+ 8n + 8n + 3 n + n + )n + ) n n n p n n n + p n 4, n, p n n n + p n 3, n. n + ) p n 8n + ) p n nn ) p n 4, n, n + 3)n + ) p n+ 8n + 8n + 3) p n 4n p n 3, n, n + ) p n p n ) nn ) p n p n 4), n n + 3)n + ) p n+ p n ) 4n p n p n 3), n Therefore, p n p n p n+ p n nn ) n + ) p n p n 4), n, ) 4n n + 3)n + ) p n p n 3), n. ) We have p, p 0, p 3, p 3, so p p 0 p p. It follows that p n, n. This gives us the desired closed form formula for p n : p n n i i + i and p n+ n+ i i +, n 0. i Defining a sequence {} for { } similar to {p n} for {p n }, we seek for {} equations corresponding to ) and ). With q 0, q 0, q, q 7 6, we have q 0, q 0, q 3, q

8 This gives us nn ) n + )... 4)) 5 q q 0) nn ) n + )... 4)) 5 9 q n+ q n 4... n) n 3, n,... n + ) n) n + 3)n + )... 5)3) q q ) n) n + 3)n + )... 5)3) n) n + 3 3, n.... n + ) Simple algebraic manipulation gives us 4... n + ) n n + ) 4... n) n 3, n,... n ) n + ) n n + ) 4... n) n + 3, n.... n ) Replacing n by i and summing over i, from to n, we find that Therefore, q n + ) n n + ), n 0, + q 4... n + ) n n + ), n n + ) n + 3, n 0,... n + ) n + ) n + 3 3, n n + ) From here we obtain the desired closed form formula for. Proof of Theorem. By Theorem 7, p n + p n n + ) n n + ), 4... n + ) n n + ). We have, Γ n + 3 ) n + n + ) Γ n + ) n + )... ) Γ ) n ) n ) Γ n ) n + n... π. 6

9 It follows that π n + )Γn + ) p n Γn + 3 π ). The above limit is due to the fact [9, 0, ] that, for any complex number a, lim n n a Γn) Γn + a), we have used a. Further + lim lim π n p n+ n p n. The reciprocal of this limit shows that HCF ) π.75. Finding HCF) The following theorem establishes closed form formulas for the numerator p n and the denominator of the convergents of HCF ). Our Theorem 4 follows from this theorem. Theorem 8 For any n 0, the numerator p n and the denominator of the convergents of HCF ) are p n n +, n + ) n i0 Proof. By the Euler-Wallis formula, the convergents pn We have ) i i + )i + ). p 0, p, p n n + p n + p n, n 0, q, q 0, n + +, n 0. n + )p n p n + n + )p n, n 0, are determined by the following recurrence relation n + )p n + np n n + )p n + n + )p n, n 0, ) n n + )p n ) n np n ) n n + )p n ) n n + )p n, n 0. Taking summation, we have Since p 0, it follows that ) n n + )p n ) n n + )p n ) p, n 0, n + )p n n + )p n + ) n p, n 0. n + )p n n + )p n, n 0, p n n + p n n + p, n 0, p n n +, n 0. 7

10 Similarly, n + ) n + ) + ) n q, n 0. Since q, it follows that n + ) n + ) + ) n, n 0, n + n + + ) n, n 0. n + )n + ) Replacing n by i and summing from i 0 to i n, we obtain n + q + n i0 ) i n i + )i + ) and this gives the desired closed form formula for. Proof of Theorem. By Theorem 8, p n n i0 i0 ) i i + )i + ) ) i, n 0, i + )i + ) So lim lim n p n n n ) i n i + )i + ) lim n i0 i0 n ) ) i lim n i + + )i+ i + i0 n ) i + lim ln. i + n i0 ) i i + i + ) The reciprocal of this limit gives HCF ) ln.59. Conclusion. We have established explicit formulas for the convergents of the first two harmonic continued fractions HCF ) and HCF ) in Theorem 3 and Theorem 4, respectively. From these formulas, we derive the values of HCF ) and HCF ) as in Theorem and Theorem. We can see that the result for HCF ) is relatively easy to establish compared to that for HCF ), as in Theorem 4 we need only one formula for any length of convergent but in Theorem 3 we have one formula for the odd-length convergents and another one for the even-length convergents of HCF ). We have not seen any result on HCF t) for a general t, but we conjecture that when t is an odd integer, we may need two formulas for odd-length and even-length convergents, and that the value of HCF t) may involve π when t is odd and involve ln when t is even. Obviously, HCF 0) and HCF t) t + t as t. Acknowledgement. The authors wish to thank the referee for many helpful suggestions. References [] C.D. Olds, Continued fractions, The Mathematical Association of America 963). 8

11 [] M. Wiener, Cryptanalysis of short RSA secret exponents, IEEE Transactions on Information Theory ) pp [3] M. Bunder and J. Tonien, A new improved attack on RSA, Proceedings of the 5th International Cryptology and Information Security Conference 06) pp [4] L. Seidel, Untersuchungen über die Konvergenz und Divergenz der Kettenbrüche, Habilschrift München 846). [5] M.A. Stern, Über die Kennzeichen der Konvergenz eines Kettenbruchs, Journal für die Reine und Angewandte Mathematik ) pp [6] L. Lorentzen and H. Waadeland, Continued fractions, Volume, Convergence theory, Atlantis Press 008). [7] A.F. Beardon and I. Short, The Seidel, Stern, Stolz and Van Vleck Theorems on continued fractions, Bulletin of the London Mathematical Society 43) 00) pp [8] L. Euler, De fractionibus continuis observationes, Commentarii Academiae Scientiarum Imperialis Petropolitanae 750) pp [9] J. S. Frame, An Approximation to the Quotient of Gamma Function, The American Mathematical Monthly 568) 949) pp [0] F.G. Tricomi and A. Erdélyi, The asymptotic expansion of a ratio of gamma functions, Pacific Journal of Mathematics ) 95) pp [] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards 964). Martin Bunder School of Mathematics and Applied Statistics, University of Wollongong, Australia mbunder@uow.edu.au Joseph Tonien Centre for Computer and Information Security Research, School of Computing and Information Technology, University of Wollongong, Australia joseph tonien@uow.edu.au 9