KAMILLA OLIVER AND HELMUT PRODINGER
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1 THE CONTINUED RACTION EXPANSION O GAUSS HYPERGEOMETRIC UNCTION AND A NEW APPLICATION TO THE TANGENT UNCTION KAMILLA OLIVER AND HELMUT PRODINGER Abstrat Starting from a formula for tan(nx in the elebrated Hakmem report [] we find a ontinued fration expansion for tan(nx in terms of tan(x Introdution Gauss hypergeometri funtion is defined by ( a b n 0 a n b n n n n! with x n x(x + (x + n a rising fatorial (An older way to write this rising fatorial is (x n ; for the hypergeometri funtions there exist different notations as well This funtion has two upper and one lower parameter and is thus often alled the two eff one Hypergeometri funtions are studied with any numbers of upper and lower parameters but this one is the most prominent one; ompare [] Hypergeometri funtions are omnipresent in mathematis and also well established in modern omputer algebra systems suh as Maple or Mathematia The following desription is borrowed from [6] Gauss hypergeometri funtion satisfies the ontiguous relation ( a b ( a b + a( b ( a + b + + ( + + To see this we ompare oeffiients of n or n 0 they math so let us assume that n We start with the right-hand side: a n (b + n a( b (a + n (b + n (a + [ ] n (b + n a(b + n a( bn ( + n n! ( + ( + n (n! ( + n n! + ( + (a + n (b + n ( + n n! ab( + n ( + an b n n n! as predited We also need a variation of this: irst we interhange a and b: ( a b ( a + b b( a ( a + b + + ( + + This author was supported by an inentive grant of the NR of South Afria
2 KAMILLA OLIVER AND HELMUT PRODINGER Now we inrease both b and by : ( a b + ( a + b (b + ( + a ( + ( + We rewrite the first ontiguous relation as ( ab ( a( b ab+ + ( + ( a+b+ + ( ab+ + and further as ( ab+ + ( ab a( b ( a+b+ + (+ ( ab+ A similar proedure applied to the variant leads to ( a+b+ + ( ab+ + This an be used in the first form: ( ab+ + ( ab (b+( a+ (+(+ (b+( a+ (+(+ + ( a+b+ +3 a( b (+ ( a+b+ + ( a+b+ +3 ( a+b+ + ( a + b Now the first form an be used again with a b replaed by a + b + + In the resulting form the variant an be used and so on The result is the ontinued fration of Gauss: ( ab+ + ( ab a( b (+ (b+( a+ (+(+ (a+( b+ (+(+3 (b+( a+ (+3(+4 The expansion is purely formal and provides more and more orret oeffiients of the powers of The book [6] disusses also the analyti validity of the expansion An appliation to tangents of multiple values Almost everybody knows that sin(x sin(x os(x sin(3x sin(x 4 os (x sin(4x sin(x 8 os3 (x 4 os(x &
3 CONTINUED RACTION EXPANSION 3 and os(x os (x os(3x 4 os 3 (x 3 os(x os(4x 8 os 4 (x 8 os (x+ & and that the polynomials that appear here are the Chebyshev polynomials Most people might have asked themselves whether there is something similar for the tangent funtion Yes there is but it is not widely known In the elebrated Hakmem report [] we find entry 6: tan(n artan(t i ( + it n ( it n ( + it n + ( it n for an integer n 0 whih is equivalent to tan(nx ( k( n k+ tan k+ (x ( k( n k tan k (x 0 k n/ 0 k n/ Compare with the sequenes [5 A A034867] This is the formula of interest; it expresses tan(nx as a rational funtion of tan(x (not a polynomial as in the simpler ases of sin(nx and os(nx or omputational (and aestethi! reasons it is however benefiial to express this rational funtion as a ontinued fration Let f( k 0 ( n k and g( ( n k k + k k 0 then we get f( g( n ( n+ ( n+ n+ 3 n+ ; this onversion into hypergeometri funtions is best done nowadays by a omputer Using Pfaff s refletion law [] ( a b ( a b ( a this an be rewritten as f( g( n ( n+ n + 3 ( n+ n
4 4 KAMILLA OLIVER AND HELMUT PRODINGER Now it is in good shape to apply Gauss ontinued fration to it: f( g( n (n + (n + 3 (n + (n (n + 3(n Observe that for natural numbers n this expansion is always finite and one does not have to worry about onvergene or our appliation we must replae by tan (x and multiply the whole expansion by tan(x The result is tan(nx n tan(x (n + (n tan (x 3 (n + (n tan (x 3 5 (n + 3(n 3 tan (x 5 7 or example tan(5x 5 tan(x 8 tan (x 7 5 tan (x 6 35 tan (x 7 tan (x 3 An independent derivation of the ontinued fration expansion Not everybody is ompletely omfortable with hypergeometri funtions hypergeometri transformations ontiguous relations et We demonstrate how suh people an also derive the ontinued fration expansion for tan(nx by using a tehnique that has produed many other beautiful expansions [3 4] We will show that f( g( a 0 + a + a +
5 with a k (4k + k + k CONTINUED RACTION EXPANSION 5 (n + j (n j This translates then readily into tan(nx a 0 and a k+ (4k + 3 a tan(x tan (x tan (x a tan (x ; k k+ (n j (n + j sine the a k s eventually beome ero this is a finite ontinued fration expansion Here is an example whih is of ourse equivalent to our previously given example: tan(x tan(5x tan (x 5 5 tan (x 8 8 tan (x 7 45 tan (x The tehnique that we use is to guess the form of the numbers a k by omputing a suffiient number of them with a omputer and deteting the pattern Of ourse one also has to give a proof and for that more guessing has to be done Define N s k ( : N N! s k+ ( : N N N! k+n k+n+ (n j N (n + j k k+n (j + (n + j N (n j k k+n+ (j + These formal power series are in fat just polynomials sine the oeffiients beome eventually ero They were also guessed using a omputer urther N s 0 ( (n j N (n + j N ( n N N! N N f( N + (j +
6 6 KAMILLA OLIVER AND HELMUT PRODINGER and N s ( N N! Now we will show that N (n + j N (n j + by distinguishing two ases: N (j + ( [ N ] s k ( a k s k ( N N! the proof that (4k + N N! k+n + k+n + k+n s k+ ( s k ( a k s k ( N k+n + (n j k k+n k+n (n + j N (n j k N N! k+n (j + ( n N g( N (n + j N (n j k + + (j + k+n (j + (n + j [ ] (n k(4k + N + (4k + (n N k (n + j N (n j k N N! k+n (j + (n + j N (n j k N (N! k+n (j + N(n + k + [ N ]s k+ (; ( [ N ] s k ( a k+ s k+ ( [ N ]s k+ ( is similar urthermore for N 0 the differenes are ero so that we get the laimed reursions (or the guessing these reursions were used to ompute a suffiient number of these polynomials
7 Consequently f( g( s 0( s ( CONTINUED RACTION EXPANSION 7 s 0 ( a 0 s 0 ( + s ( a 0 + s ( s 0 ( a 0 + a + s ( s ( whih leads to the promised ontinued fration Notie again that the proess stops sine n is a non-negative integer Referenes [] M Beeler R W Gosper and R Shroeppel HAKMEM MIT AI Memo 39 online version at hbaker/hakmem/hakmemhtml 97 [] R Graham D E Knuth and O Pathasnik Conrete Mathematis seond edition Addison Wesley Reading 994 [3] N S S Gu and H Prodinger On some ontinued fration expansions of the Rogers-Ramanujan type Ramanujan Journal 6: [4] K Oliver and H Prodinger Continued frations related to Göllnit little partition theorem Afrika Matematia 0 [5] N J A Sloane The online enylopedia of integer sequenes Online version at wwwresearhattom/ njas/sequenes/ [6] H S Wall Analyti Theory of Continued rations Chelsea New York Erlangen Germany address: olikamilla@gmailom Department of Mathematis University of Stellenbosh 760 Stellenbosh South Afria address: hproding@sunaa
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