* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download l - OPUS at UTS - University of Technology Sydney
Survey
Document related concepts
Mathematical proof wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Georg Cantor's first set theory article wikipedia , lookup
Ethnomathematics wikipedia , lookup
History of mathematics wikipedia , lookup
Foundations of mathematics wikipedia , lookup
Hyperreal number wikipedia , lookup
List of important publications in mathematics wikipedia , lookup
Fundamental theorem of algebra wikipedia , lookup
Series (mathematics) wikipedia , lookup
Elementary mathematics wikipedia , lookup
Collatz conjecture wikipedia , lookup
Transcript
INVERSE TRIGONOMETRIC AND HYPERBOLIC SUMMATION FORMULAS INVOLVING GENERALIZED FIBONACCI NUMBERS R. S. Melham and A. G. Shannon University of Technology, Sydney, 2007, Australia (Submitted June 1993) 1. INTRODUCTION Define the sequences {Un}™=0 and {Vn}^0 for all integers n by U„ = pU„_, + U„_2 ,UQ=0, Vn = Uy = \,n>2, pVn_l+Vn_2,V^2,Vl=p,n>2. (1.1) (1.2) Of course, these sequences can be extended to negative subscripts by the use of (1.1) and (1.2). The Binet forms for U„ and V„ are a"-pn a-p (1.3) V„ = a"+P", (1.4) t/„ = and where a = p + Jp2+4 2 /7--y//+4 ' P 2 (1.5) Certain specializations of the parameter/? produce sequences that are of interest here and Table 1 summarizes these. TABLE 1 p 1 2 u„ F„ K L„ 2x Pn{*) Qn Q„{*) Pn Here {Fn} and {Ln} are the Fibonacci and Lucas sequences, respectively. The sequences {Pn} and {Qn} are the Pell and Pell-Lucas numbers, respectively, and appear, for example, in [3], [5], [7], [11], and [17]. The sequences {P„(x)} and {Qn{x)} are the Pell and Pell-Lucas polynomials, respectively, and have been studied, for example, in [12], [14], [15], and [16]. Hoggatt and Ruggles [9] produced some summation identities for Fibonacci and Lucas numbers involving the arctan function. Their results are of the same type as the striking result of D. H. Lehmer, Stan"1 7=1 32 K (1.6) V^I+1 [FEB. INVERSE TRIGONOMETRIC AND HYPERBOLIC SUMMATION FORMULAS to which reference is made in their above-mentioned article [9]. Mahon and Horadam [13] produced identities for Pell and Pell-Lucas polynomials leading to summation formulas for Pell and Pell-Lucas numbers similar to (1.6). For example, X t a n ~ x \^2i+lJ K (1.7) ~2' 7=0 Here, we produce similar results involving the arctan function and terms from the sequences {£/„} and {Vn}. Some of our results are equivalent to those obtained in [13] but most are new. We also obtain results involving the arctanh function, all of which we believe are new. 2. PRELIMINARY RESULTS We make consistent use of the following results which appear in [1] and [6]: tan *x + tan l y- tan tan l x - tan l y = tan l L if xy <1, l-xy l (2.1) f x-y} if xy > - 1 , l + xy, (2.2) tanh 1 x + tanh ^ ^ t a n h l\ x + y l + xy (2.3) tanh -1 x - tanh-1 y = tanh"1 (2.4) tanhi - i x = 1—, log.( 11 -+ x 2 \l coth l x = —log. 2 6 l-xy |JC|<L I, |x|> 1, (2.5) (2.6) %x-1 tan l x = cot l tanh x x = coth * (2.7) (2.8) We note from (2.7) and (2.8) that all results obtained involving arctan (arctanh) can be expressed equivalently using arccot (arccoth). If k and n are integers, and writing A = (a-/?r = / / + 4 (2.9) we also have the following: 19951 33 INVERSE TRIGONOMETRIC AND HYPERBOLIC SUMMATION FORMULAS u2„-un+ku„_k = (-iy+ku2k, (2.io) K+kv„-k-vn2 = H-irku2k, (2.ii) k even, (ULV-, un+k-u„_k= ;;• ' lt/„r t , A: odd, Un+k + U^k= (U„VL., "*' A: even, (2.i2) ' (2.13) lt4F„, £odd, ^*-^t= fAf/tC/„, * T t F„, A: even, , ' A: odd, (2-14) ^ + r ^ = fVkV„, *"' [A^f/„, A: even, , ' /todd. (2.15) ^ 2 + ^ = ^ 1 , (2-16) UnU^H-W = Ulv (2.17) Identities (2.12)-(2.15) occur as (56)-(63) in [2], and the remainder can be proved using Binet forms. Indeed, (2.10) and (2.11) resemble the famous Catalan identity for Fibonacci numbers, Fn2-F„,kFn_k=(-irkFk2. (2.18) We assume throughout that the parameter/? is real and \p\ > 1. If p > 1, then {Un}™=2 and iK)n=i a r e increasing sequences. If p < - 1 , then {\Un\}™=2 and {|FJ}*=1 are increasing sequences and, if n>0, then Un < 0, U„>0, F„ < 0, F„ > 0, n even, n odd, n odd, # even. (2.19) Furthermore, if \p\ > 1, then using Binet forms it is seen that ,. U„+m ,. V' Sm, lim_JM,lim!mJ ' „_>«, £/w «^co ^ l_^? mevenor/?>l, ^ ' OT odd and/? ^ - 1 , (220) where o 34 I/ ? I+A// ? 2 +4 ' [FEB. INVERSE TRIGONOMETRIC AND HYPERBOLIC SUMMATION FORMULAS 3, MAIN RESULTS Theorem 1: Tfn is an integer, then tan lUn+2~tm f i A tan yUnJ l „ A ( l U„=tm n even, (3.1) n odd, « ^ - l . (3.2) \Pn+\) 1 A f + tan - tan -l f TT \ n+l v. , \Un+lJ \&n+2j Proof: f tan * [/„, 'n+2 0 ~ tan * [/„ = tan K1 + U U » »+2J = tan „ \ \Un+lJ where we have used (2.2), (1.1), and (2.17). To prove (3.2), proceed similarly using (2.1), (2.16), and (2.17). • Now, in (3.1), replacing n by 0, 2, ..., In - 2, we obtain a sum which telescopes to yield I tan"1 7= 1 Uu-i = tan_1£/,In- (3.3) Similarly, in (3.2), replacing n by 1, 3, ..., In - 1 yields (tr \ 7T _1 + - 1 = - + (-lf\«-l tan4 V^2n+lJ v^-y I(-l)'-1tan /=! (3.4) The corresponding limiting sums are Ztan"-i P U 2i-i y />*i, -p (3.5) PS-1. Z(-D"tan n /=! ~4' (3.6) We note here that (3.3) and (3.4) were essentially obtained by Mahon and Horadam [13], (3.3) in a slightly different form. When p = l, (3.5) reduces essentially to Lehmer's result (1.6) stated earlier. Theorem 2: For positive integers k and n, tan '_0 Un-k t -tan KU n -l Proof: Use (2.2), (2.10), and (2.13). D 1995] tan _ tan -l . VkU2n j a2« |, k even, (3.7) A: odd. J 35 INVERSE TRIGONOMETRIC AND HYPERBOLIC SUMMATION FORMULAS Now, in (3.7), replacing n by k, 2k, ..., nk to form a telescoping sum yields I tan"1 U,nk = tan \ /=! u U Stan l V U2ik 7=1 j A even (3.8) ;, k odd. (3.9) {rl+l)kj = tan nk U \ {n+\)k J Using (2.20), the limiting sums are, respectively, E t a n -I 7=1 Itan"1 7= 1 f \ TT2 Ui* VJJl kuik - ftan o « - V(S > r * ), ^(-ly- 1 ^"! = JtaiT'OT*), v (3.10) £even, y ^2* * odd, p>\, tan" 1 ^ ), Arodd, y (3.11) p<-\. Theorem 3: For positive integers k and n, tan K^ ^ -tan -if ^ -l k even, (3.12) T,77+fc . V^. A(-iTO2 , tan A: odd. tan V ^2" J Proof: Use (2.2), (2.11), and (2.15). • Again in (3.12), replacing n by k, 2k, ..., nk yields \ A( V. 2> " | | = «ar'[|-l-«a n -V^(77+i)/vy , n ;=1 nk W, £ tairl f(zirV = tan K2/jt 7=1 -tan y */ £even, K 77/C kodd. (3.13) (3.14) V^(77+l)/:y Since the left sides of (3.9) and (3.14) are the same, we can write tan U,nk \U(n+l)kJ + tan Knk - tan l\ — J, A:odd, (3.15) V (n+l)k) and taking limits using (2.20) gives t a n - 1 ( 0 = ^ t a n - 1 f | | | j , kodd. The limiting sum arising from (3.13) is 36 (3.16) INVERSE TRIGONOMETRIC AND HYPERBOLIC SUMMATION FORMULAS f o A Stan_1 = tan tan \S~K\ k even. \nj (3.17) Theorem 4: If?2>2, then f v\ tanh - l tanh + tanh * l = tanh * Kw+1 2 #.n+2 V^«+l -tanh \UnJ ' = tanh * ft even, (3.18) , n odd. (3.19) \Un+2j Proof: To prove (3.18) use (2.3), (2.16), and (2.17); (3.19) is proved similarly. • These results lead, respectively, to XC-iy^tanh-1 Vim Z= \U2M l = tanh 1' l ^ + (-l)" _1 tanh- 1 \U<j J f 1 ^ (3.20) \L>2n+4 J A ( 1 ^ < \^ (3.21) tanh l ^ t a n h -l - tanh- l \U3; \^2n+3 J \^2i+2 J /=! Note that in Theorem 4 our assumption that w > 2, together with our earlier assumption that \p\ > 1, is necessary to ensure that the arctanh function is defined. The corresponding limiting sums are ^(-ly^tanh- 1 = tanh * J_ (3.22) yU2i+3j Vtanh-f-^-Vtanh-1 'O (3.23) \U3J We refrain from giving proofs for the theorems that follow, since the proofs are similar to those already given. Theorem 5: Let n > k be positive integers where (k, n) *• (1,1) ifp - ±1. Then tanh ' tanh 1 -tanh u-n-k -l \Un J k even, V U2n (3.24) 1 tanh" v vku\ j k odd. This leads to Z/=i 1995] tanh r U^ \U2ik - tanh" j Unk t , k even, (3.25) \U(n+l)Jc J 37 INVERSE TRIGONOMETRIC AND HYPERBOLIC SUMMATION FORMULAS Jtanh"1 7= 1 v vku% j U,nk - tanh k odd. (3.26) \U(n+\)k J The corresponding limiting sums are f>nh-! U^ f = tanh \d k), f 2] tanh -1 v vkufk j 7=1 k even, (3.27) \Plik J 7=1 f tanh_1(<r*), A: odd, p>\, -1 k odd, p<~\. )-tanh 0T*)> (3.28) Theorem 6: Let n > k be positive integers where (k, n) & (1,1) if 1 < \p\ < 2. Then tanh r -1 n-k KK f -1 V -tanh tanh - 1 ^ \Yn+k J J tanh l (-iro U2„ k even, j (3.29) A(-l)-^r 2 VkV A: odd. j n The resulting sums are = tanh 1 £ tanh" \Unk j 7=1 Jtanh"1 7=1 -tanh k even, \V(n+\)k yVkj vkvl j - tanh - 1 f n \ KVkj tanh - 1 (3.30) J Knk \V{n+\)k &odd. (3.31) J Since the left sides of (3.25) and (3.30) are the same, we can write tanh 1 U,nk + tanh - l -l f o "* i, k even, (3.32) -l k even. tanh-'(<T*) = -tanh yVkj 2 This should be compared with (3.16). The limiting sum arising from (3.31) is (3.33) \U{n+l)k j = tanh \Vkj and taking limits yields ( *> \ 1 Xtanh' 7= 1 1' 1 A(-l)'- ^ vkvl j tanh"1 <1^ tanh"1^), kodd, p>\, (3.34) tanh" _1 KKJ + tanh (<T*), kodd, p<-\. At this point we remark that Mahon and Horadam [13] obtained results similar to our Theorems 2 and 3 and derived summation formulas from them. However, in our notation, they considered only the case k odd. 38 [FEB. INVERSE TRIGONOMETRIC AND HYPERBOLIC SUMMATION FORMULAS 4 APPLICATIONS We now use some of our results to obtain identities for the Fibonacci and Lucas numbers. From (3.22) and (3.23), we have XC-iy^tanh" 1 L'2/+3 2 77 7= 1 4 log * 2 ' 1 log, 3. Xtanh' 1 (4.1) (4.2) \*'2i+2 J 1=1 In terms of infinite products, these become, respectively, r r -^2/+3 + (~ v F /=1 Az/+3 _ o (4.3) l lL 2i+3+(- ) 2i+3 n °° ^2/+2 + 1 _ 3 F -1 /=1 ^2/4-2 (4.4) l In (3.28), keeping in mind the constraints in the statement of Theorem 5 and taking k - 3, we obtain f 1 A (4.5) XC-iy-'tanh', - 1 F = - l o g , </>, \ l) i=i or (4.6) where <j> = —p- is the Golden Ratio. Finally, (3.34) yields, after simplifying the right side, ^(-l)'-1tanh_1 ' 5 ^ /=i \L*iJ = -log e (3(^-l)), (4.7) or U4+(-i>'5 (4.8) Of course, many other examples can be given by varying the parameter k. REFERENCES 1. M. Abramowitz, & I. A. Stegun. Handbook of Mathematical Functions. New York: Dover, 1972. 2. G. E. Bergum & V. E. Hoggatt, Jr. "Sums and Products for Recurring Sequences." The Fibonacci Quarterly 13.2 (1975): 115-20. 3. M. Bicknell. "A Primer on the Pell Sequence and Related Sequences." The Fibonacci Quarterly 13.4 (1975):345-49. 1995] 39 INVERSE TRIGONOMETRIC AND HYPERBOLIC SUMMATION FORMULAS 4. O. Brugia, A. Di Porto, & P. Filipponi. "On Certain Fibonacci-Type Sums." International Journal ofMathematical Education in Science and Technology 22 (1991):609-13. 5. E. Cohn. "Pell Number Triples." The Fibonacci Quarterly 10.5 (1972):403-12. 6. C. H. Edwards, Jr., & D. E. Penny. Calculus and Analytic Geometry. Englewood Cliffs, NJ: Prentice Hall, 1982. 7. J. Ercolano. "Matrix Generators of Pell Sequences." The Fibonacci Quarterly 11.1 (1979): 71-77. 8. V. E. Hoggatt, Jr., & I. D. Ruggles. "A Primer for the Fibonacci Numbers—Part IV." The Fibonacci Quarterly 1.4 (1963):65-71. 9. V. E. Hoggatt, Jr , & I. D. Ruggles. "A Primer for the Fibonacci Numbers—Part V." The Fibonacci Quarterly 2.1 (1964):59-65. 10. A. F. Horadam. "Generating Identities for Generalized Fibonacci and Lucas Triples." The Fibonacci Quarterly 15.4 (1977):289-93. 11. A. F. Horadam. "Pell Identities." The Fibonacci Quarterly 9.3 (197rl):245-52, 263. 12. A. F. Horadam & Bro. J. M. Mahon. "Pell and Pell-Lucas Polynomials." The Fibonacci Quarterly 2X1 (1985):7-20. 13. Bro. J. M. Mahon & A. F. Horadam. "Inverse Trigonometrical Summation Formulas Involving Pell Polynomials." The Fibonacci Quarterly 23.4 (1985):319-24. 14. Bro. J. M. Mahon & A. F. Horadam. "Matrix and Other Summation Techniques for Pell Polynomials." The Fibonacci Quarterly 24.4 (1986):290-309. 15. Bro. J. M. Mahon & A. F. Horadam. "Ordinary Generating Functions for Pell Polynomials." The Fibonacci Quarterly 25.1 (1987):45-56. 16. Bro. J. M. Mahon & A. F. Horadam. "Pell Polynomial Matrices." The Fibonacci Quarterly 25.1 (1987):21-28. 17. C. Serkland. "Generating Identities for Pell Triples." The Fibonacci Quarterly 12.2 (197r4): 121-28. AMS Classification Numbers: 11B37, 11B39 Mark Your Calendars Now! Mathematics Awareness Week 1995 "Mathematics and Symmetry" April 23-29 1995 Every year, Mathematics Awareness Week celebrates the richness and relevance of mathematics and provides an excellent opportunity to convey this message through local events. During a week-long celebration from Sunday, 23 April - Saturday, 29 April 1995, the festivities will highlight "MATHEMATICS AND SYMMETRY". Mark your calendars now and plan to observe Mathematics Awareness Week in your area, school, or organization. Look for further information from the Joint Policy Board for Mathematics, national sponsor of Mathematics Awareness Week, in future issues of The Fibonacci Quarterly. 40 [FEB. The Fibonacci Quarterly Journal Home | Editorial Board | List of Issues How to Subscribe | General Index | Fibonacci Association Volume 33 Number 1 February 1995 CONTENTS Cover Page Referees Full text 2 Clark Kimberling The Zeckendorf Array Equals the Wythoff Array Full text 3 W. R. Spickerman, R. L. Creech, and R. N. Joyner On the (3, F) Generalizations of the Fibonacci Sequence Full text 9 R. S. Melham and A. G. Shannon Some Infinite Series Summations Using Power Series Evaluated at a Matrix Full text 13 Julia Abrahams Varn Codes and Generalized Fibonacci Trees Full text 21 Roger Webster A Combinatorial Problem with a Fibonacci Solution Full text 26 R. S. Melham and A. G. Shannon Inverse Trigonometric Hyperbolic Summation Formulas Involving Generalized Fibonacci Numbers Full text Morris Jack DeLeon Sequences Related to an Infinite Product Expansion for the Square Root and Cube Root Functions Full text Marjorie Bicknell-Johnson and David A. Englund Greatest Integer Identities for Generalized Fibonacci Sequences {H n }, Where H n = H n-1 + H n-2 Full text http://www.fq.math.ca/33-1.html[30/07/2012 11:48:32 AM] 32 41 50 The Fibonacci Quarterly Wayne L. McDaniel Diophantine Representation of Lucas Sequences Full text 59 R. S. Melham and A. G. Shannon Some Summation Identities Using Generalized Q-Matrices Full text 64 Vassil S. Dimitrov and Borislav D. Donevsky Faster Multiplication of Medium Large Numbers Via the Zeckendorf Representation Full text 74 Paul Thomas Young Quadratic Reciprocity Via Lucas Sequences Full text 78 R. S. Melham and A. G. Shannon A Generalization of the Catalan Identity and Some Consequences Full text 82 Edited by Stanley Rabinowitz Elementary Problems and Solutions Full text 85 Edited by Raymond E. Whitney Advanced Problems and Solutions Full text 91 Back Cover Copyright © 2010 The Fibonacci Association. All rights reserved. http://www.fq.math.ca/33-1.html[30/07/2012 11:48:32 AM]