Research Article Integer Powers of Arcsin

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1 Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 007 Article ID pages doi:0.55/007/938 Research Article Integer Powers of Arcsin Jonathan M. Borwein and Marc Chamberland Received November 006; Revised 8 February 007; Accepted 8 April 007 Recommended by Ahmed Zayed New simple nested-sum representations for powers of the arcsin function are given. This generalization of Ramanujan s wor maes connections to finite binomial sums and polylogarithms. Copyright 007 J. M. Borwein and M. Chamberland. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original wor is properly cited.. Introduction This note discovers derives and then studies simple closed-form Taylor series epressions for integer powers of arcsin(. Specifically we show that for and N... where H ( /4and H N+ (: 4 arcsin N (/ (N! n and also that for andn 0... arcsin N+ (/ (N +! H N ( (. ( n ( ( n n n n N n N ( (. nn G N ( ( ( +4 + (.3 0

2 International Journal of Mathematics and Mathematical Sciences where G 0 ( and G N (: n 0 ( n + n n 0 n N ( n + ( nn +. (.4 The convention is that the sum is zero if the starting inde eceeds the finishing inde. Nested sums are not new. The last decade saw many interesting results concerning Euler sums or multizeta values wonderful generalizations of the classical ζ-function whose discovery can be traced to a letter from Goldbach to Euler [ pages 99-00] and [Chapter 3] a letter that played a seminal role in the discovery of the ζ-function. When Gauss was criticized for the lac of motivation in his writings he remared that the architects of great cathedrals do not obscure the beauty of their wor by leaving the scaffolding in place after the construction has been completed. While we find (. and (.3 worthy of undistracted attention in truth their discovery was greatly facilitated by the use of eperimental mathematics the relatively new approach to doing mathematical research with the intelligent use of computers. This perspective is elucidated throughout this paper. It is also illustrative of the changing speed of mathematical communication that the special cases (. (. and (.3 given below are already online at [3]. n N 0. Eperiments and proofs The first identity below is very well nown: ( arcsin (. (. It is eplored at some length in [4 pages ]. While it is seen in various calculus boos (see [5 pages 88 90] where the series for arcsin 3 ( isalsoprovenitdates bac to at least two centuries and was given by Ramanujan among many others; see [6 pages 6-63]. As often in Mathematics history is complicated. Equation (. has been rediscovered repeatedly. For eample an equivalent form is elegantly solved as a 96 MAA Monthly problem [7 Problem E 509 Page 3]. We quote in etenso the editors attempts to trace the history of the formula: The series was located in the Smithsonian Mathematical Formulae and Tables of Elliptic Functions 6.4 No. 5 p. ; Chrystal Algebra vol ed E. No. 7 p. 335 (cites Pfaff as source; Bromwich An Introduction to the Theory of Infinite Series 908 ed Prob. page 97 (claims nown to Euler; Knopp Theory and Application of Infinite Series E. 3 Chap. VIII p.7; Schuh Leerboe der Differtiaal en Integraalreening vol. pp ; Hobson Treatise on Plane Trigonometry eqs. 0 pp ; M.R. Speigel this Monthly 60 ( ; Taylor Advanced Calculusp.63; EdwardsDifferential Calculus for Beginners (899 p.78. Note the appearance in the Monthly itself in 953.

3 J. M. Borwein and M. Chamberland 3 The second identity slightly rewritten (see [6] is less well nown ( arcsin 4 3 { m m } (. and when compared they hint at the third and fourth identities below subsequently confirmed numerically the prior flimsy pattern and numerical eploration suggest that arcsin 6 ( arcsin 8 ( { m m m n { m m m n n ( } n ( n p } p Reassured by this confirmation we conjectured that in general arcsin N (/ (N! (. (.3 H N ( (.4 where H N+ ( isasin(.. Subsequently we were naturally led to discover the corresponding odd formulae. We net provide a proof of both (. equivalently (.4 and (.3. Proof of (.and(.3. The formulae for arcsin (with 4aregivenin[6pages 6-63] and Berndt comments that [5] is the best source he nows for and3. Berndt s proof also implicitly gives our desired result since he establishes via a differential equation argument that for all real parameters a one has where e aarcsin( n0 ( c n (a n n! (.5 n ( c n+ (a: a a +( n ( c n (a: a +(. (.6 Now epanding the power of a n on each side of (.5 provides the asserted formula. Note that (.5 is equivalent to the somewhat-less-elegant if more-concise [8 Formula ] specifically [ ( ] (ia / ( i ep asin. (.7!(ia + / 0

4 4 International Journal of Mathematics and Mathematical Sciences Another proof can be obtained from the hypergeometric identity ( sin(a +a asin( F a ; 3 ;sin ( givenin [4 Eercise 6 page 89]. Maple can also prove identitiessuch as (.5 as the following code shows: > ce:n->product(a^+(*^0..n-: > co:n->a*product(a^+(*+^0..n-: > sum(ce(n*^(*n/(*n!n0..infinity assuming >0; cosh(a arcsin( (.8 > simplify(epand(sum(co(n*^(*n+/(*n+!n0..infinity assuming >0; sinh(a arcsin( A necessarily equivalent formula for powers of arcsin is listed by Hansen in [8Formula 88..] m n n0 { n n 0 ( } n n! [( ] ( n n! n n + (n n n m! (n m! (n m + nm ( sin m (.9 but its relation to (. and(.3 is not obvious; our nested-sum representations are definitely more elegant. While Ramanujan listed only the first two cases in his noteboos [6 pages 6-63] his previous entries suggest that an approach for any power was being assembled. One can only imagine how much farther his intuition would have taen him if he had today s computational power! Powers of arcsin play an important role in analytical calculations of massive Feynman diagrams see [9] and in the construction of Laurent epansions of different types of hypergeometric functions with respect to small parameters. In [9] series epansions for small powers of arcsin are given but they do not have the compact nested-sum form seen in (. and(.3. Liewise formula (.5 has proven central to recent wor on effective asymptotic epansions for Laguerre polynomials and Bessel functions [0]. These results are again motivated largely by applications in mathematical physics and in prime computation. 3. Properties of coefficient functions It is of some independent interest at least to the present authors to determine a few properties of H N (andg N (. Corollary 3.. The following properties obtain: (a H N ( G N ( 0 if N>;

5 J. M. Borwein and M. Chamberland 5 (b H N ( jn (H N (j/(j G N ( jn (G N (j/(j + ; (c H ( /4 (! G ( 4! /(! ; (d N ( 4 N H N ( 0 for N0 ( N G N ( 0 for. Proof. Parts (a (c follow from the definition of H N ( andg N (. To prove part (d first note that (.may be rewritten as Use this to obtain N (N! 4 N 4 +N H N ( (sin. (3. ( sin cos( + + N ( N 4 N N (N! ( N 4 +N H N ( (sin N ( ( (sin ( ( 4 N H N (. Now matching powers of (sin implies the first part of (d. Equation (.3 maybe similarly manipulated to give the second part of (d. Correspondingly we have the following corollary. Corollary 3.. For nonnegative integers N and one has ( ( 4 j j H N ( j j G N ( j (3.3 j ( ( GN ( ( Proof. Differentiate (.3to obtain j0 j0 ( j j 4 j N ( j(j ( j j ( ( arcsin N (N! 0 H N+ ( j. (3.4 G N ( ( 6. (3.5 Using the binomial theorem and comparing to (.gives(3.3. Similarly differentiating (.leadsto(3.4. Even the simplest case with N yields the nonobvious identity ( ( 4 j j j j j j ( j0 n0 (n +. (3.6

6 6 International Journal of Mathematics and Mathematical Sciences Asing a student to prove this identity directly is instructive particularly from an eperimentalmathematicalperspective. WhileMaple stares dumbly at the right-hand side it immediately redeems itself after one interchanges the order of summation producing n0 (n + n j0 ( j j ( j j j π4 Γ( 4Γ( +/ (3.7 which may be easily rewritten as the desired epression (3.6. A host of partition identities tumble directly from (.3and(.4 on comparing various ways of combining powers of arcsin. Corollary 3.3. Given m n {M i } m i and{n i } n i 0let T : M + +M m + ( N + + +(N n + W : n( M! ( Mm! ( N +! (N n +!. (3.8 Then for any s nwheres and n have the same parity ( ( W n ( ( ( ( HM j HMm jm GN GNn n n 4 n ( ( j jm (j ( j jm j m ( n 4 + +n 4T! H ( T/(s/ s s/ s n even ( ( T! s G(T / (s / s s4 s n odd where the sum is taen over all partitions of (3.9 s + n j + + j m n (3.0 where j j... j m and... n. An interesting special case occurs when we specify n 0 m M M and s : j ( j j ( j j j ( j 6 ( j j (3. hence lim j ( j j ( ( j j j ( j π. (3.

7 J. M. Borwein and M. Chamberland 7 4. Powers of arcsin via iterated integrals Since arcsin ( / we may also epress any power of arcsin( asaniterated integral. Specifically arcsin n n! 0 dy y y 0 dy y y 0 dy yn 3 y3 0 dy n y n. (4. To convert the multiple integral into a multiple sum note that the binomial theorem gives 4 0 which may be repeatedly used in (4.toyield arcsin n n!... n0 { n 0 ( ( ( n n w n+n ( 4 wn + ( + + (w n + n 0 ( 4 ( + { n ( ( 0 ( (4. ( 4 ( + + ( ( + n n + n w n n0 ( n n 4 n ( wn + n ( n n ( n n n + n } 4 n } (4.3 where w n : n. Though this process also writes the coefficients in terms of nested sums these terms are not nearly as simple as H N and G N. 5. Related series manipulations After having discovered formulae such as (.and(.3 the analyst s natural inclination is to mine them for striing eamples. We rescale (. and(.3 to obtain arcsin N H ( (N! N (4 ( G arcsin N+ N ( ( ( (N +! ( (5. These series or their derivatives may be evaluated at values such as π/ π/3 π/4 π/6 i/ and i to obtain many formulae along the lines of those found by Lehmer and

8 8 International Journal of Mathematics and Mathematical Sciences othersas described in [4 pages ]. A few eamples are H N (4 ( ( π N (N! H N (3 ( ( π N (N! 3 H N ( ( ( π N (N! 4 H N (( ( H N ( ( π N (N! 6 ( ( ( N log (N! 5 N (5. H N (( 4 ( ( N ( ( N. log + (N! Integrating (. and(.3 is naturally much more challenging. Replacing by i in (. and(.3 provides the Maclaurin series for positive integer powers of the form log N ( + +. These epressions may then be integrated (or differentiated. In particular see [4Eercise 7 page 89] we have ( ( n+ 4 n n 3 n n / 0 log ( + + d ζ(3 0. (5.3 Liewise with more wor see [] we obtain / ln 4 ( + + d 3 Li 5( g ( +3Li 4 g ln(g+ 3 ζ(5 0 5 ζ(3ln (g 4 5 π ln 3 (g+ 4 5 ln5 (g (5.4 where g ( 5 / is the golden ratio and where Li n (z z / n is the polylogarithm of order n. This process may be generalized as follows. Define / n : n ln n ( + + d. (5.5 n! 0 We leave it to the reader to determine the eplicit series epression for n forn is even and for n is odd. Etensive eperimentation with Maple coupled with pattern looups with Sloane s online encyclopedia []produced n ζ(n + n( logg n+ (n +! n+ ( logg n+ j ( Li j g. (5.6 (n + j! j

9 J. M. Borwein and M. Chamberland 9 Multiplying by n and summing over n shows that this is equivalent to the net generating function written in terms of ψ the digamma function: logg 0 ( e y coth(ydy + ( γ + ψ( /+logg e logg + e logg ( Li g. (5.7 These observations can be made rigorous with the net result. Theorem 5.. For n and n! 0 arcsin n (y y n+ dy Li (( + i ( i arcsin( n+ (n +! iarcsinn+ ( (n +! + arcsinn ( log ( i ( n! i n + ζ(n +. (5.8 Proof. The derivatives of each side match and the equation holds for 0. Alternatively this is a reworing of [3 Formula (7.48 page 99]. In hindsight this equation should not come as much of a surprise since Ramanujan s entries immediately preceding (. give similar formulae using the Clausen functions. One can substitute (. or(.3 to obtain further formulae. This allows for etensions to the comple plane. For eample we have the following corollary. Corollary 5.. For n andre( 0 ( n H n (( 4 3 ( n+ Eample 5.3. Substituting i gives Li (( + arcsinh( n+ (n +! + arcsinhn+ ( (n +! + ζ(n + 4 n. + arcsinhn ( log ( + n! (5.9 H n (4 n 3 ( ( ( ζ( + (π/ n 4 (n! + (π/n (n! ( log + n ζ(n + 4 (5.0

10 0 International Journal of Mathematics and Mathematical Sciences with the special case n yielding 6. Conclusion 4 ( 3 π log 7 ζ(3. (5. We hope this eaminationof (.and(.3 encourages readers to similarly eplore what transpires for arctan and other functions. Armed with a good computer algebra system and an internet connection one can quite fearlessly undertae this tas. Acnowledgment Research is supported by NSERC and by the Canada Research Chair Programme. References [] J.M.BorweinandD.BaileyMathematics by Eperiment A K Peters Natic Mass USA 004. [] J. M. Borwein D. Bailey and R. Girgensohn Eperimentation in MathematicsAKPetersNatic Mass USA 004. [3] E. W. Weisstein Inverse Sine from MathWorld-A Wolfram Web Resource. [4] J.M.BorweinandP.B.BorweinPi and the AGM: A Study in Analytic Number Theory and Computational Compleity Canadian Mathematical Society Series of Monographs and Advanced Tets John Wiley & Sons New Yor NY USA 987. [5] J. Edwards Differential Calculus MacMillan London UK nd edition 98. [6] B. C. BerndtRamanujan s Noteboos Part I Springer New Yor NY USA 985. [7] A. G. Konheim J. W. Wrench Jr. and M. S. Klamin A Well-Known Series American Mathematical Monthly vol. 69 no. 0 p. 0 December 96. [8] E.R.HansenA Table of Series and Products Prentice-Hall Englewood Cliffs NJ USA 975. [9] M. Yu. Kalmyov and A. Sheplyaov lsj a C++ library for arbitrary-precision numeric evaluation of the generalized log-sine functions Computer Physics Communications vol. 7 no. pp [0] D. Borwein J. M. Borwein and R. E. Crandall Effective laguerre asymptotics submitted to SIAM Journal on Numerical Analysis. [] J. M. Borwein D. J. Broadhurst and J. Kamnitzer Central binomial sums multiple Clausen values and zeta values Eperimental Mathematics vol. 0 no. pp [] N. J. Sloane Online Encyclopedia of Integer Sequences ( njas/sequences/. [3] L. Lewin Polylogarithms and Associated Functions North-Holland New Yor NY USA 98. Jonathan M. Borwein: Faculty of Computer Science Dalhousie University Halifa NS Canada B3HW5 address: jborwein@cs.dal.ca Marc Chamberland: Department of Mathematics and Statistics Grinnell College Grinnell IA 50 USA address: chamberl@math.grinnell.edu