A SOLUTION TO A TANTALIZING PROBLEM. GERT ALMKVIST Institute for Algebraic Meditation, PL 500, S HOOR, Sweden (Submitted November 1984)

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1 GERT ALMKVIST Istitute for Algebraic Meditatio, PL 500, S HOOR, Swede (Submitted November 984) INTRODUCTION I a recet paper, R. Backstrom [] computed various sums of reciprocal Fiboacci ad Lucas umbers. By a strage limit process, he also gets a estimate (to the seveth decimal place) of the sum ^, \/5 + ^ 7 T~^ ~ "5" + 7~» where a = ~ ~ L log a (here ^ L 0 =,' L, =, ' ad L «= L -, + - ). A eve better estimate is the / formula ^ 0 L X ' «+ X l 0 g a 7I ' - (log a ) gtt /log a _ ' which has at least thirty correct decimal places. But both these formulas are just the first terms i a very rapidly covergig series, that is, a quotiet of two theta fuctios. This paper cotais o ew results. O the cotrary, most of the results are approximately 50 years old, mostly due to Jacobi. The formulas for the sums of reciprocal Fiboacci ad Lucas umbers are obtaied by substitutig q = a - or q = a~ i idetities valid for formal power series or for series covergig for \q\ <. Probably all the results i Backstrom*s paper ca be obtaied by specializig to q = a - or q = a" i sums of telescopig series. For example, let us look at Theorem I i []: j,,, + g -*i(* +?)K+i = 0 x + l r + l l We have i F + + F r + S L r + l \ l + a-( + r+l) { + a - ( - r ) L r + l \ l + q + r + l + q-r, where q Hece, i t i s s u f f i c i e t to show t h a t l - 0 \ l + q + r + l j + qr, * = r +-j for 0 < \q\ <. Now, 36 [Nov.

2 E - E = 0 \l + q +r+ + q - r / v = N-r + i l + qv V = _ P x + q x r + l - f p + j = p + y as / -»- 9 + g v + ^ " for V + 0. Here we ever used the fact that q = a~, so the summatio of the ier series has othig to do with Fiboacci umbers. We hope to get some of the Fiboacci ethusiasts iterested i theta fuctios. A excellet text is Rademacher f s lecture otes [6]. They pair Germa thoroughess with elegace. O the other side of the spectrum is Bellma f s very thi book [], which cotais almost o proofs s oly the most importat results ad some applicatios.. THETA FUNCTIONS We have the followig theta fuctios (the summatio is over all i 7L) : \{X, q) =l (-l)y( + (l/)]y(, + l). \(.x, q) = q [«+ (/)] gi(+l)ttx * 3 (*, q) = L q e i ; \(x, q) = E (-DVV *. () We make the s u b s t i t u t i o q = e^iz ad get the followig fuctioal equatios: ad >(! -i)-w? *-' "-<*» (/I is take i the first quadrat*) This was essetially proved i 83 by Poisso i the form: + e~ + e~^x + e~ VS + e^,x + e~^/x + e~ 9lj/x +... Notatio: I the sequel we will oly have to cosider the case x = 0. We will write»?>-(&)-.(o.,>. 986] 37

3 We have may formulas, as follows: & = Tf ( ~ l ) ( + l) q l+(l/)l. d =? t + ( l / ) ] =. \ - Z i-l) q \» k (-q) «d 3 (q). #'" = -TT 3 ( - l ) " ( w + ) V «+ a / ) ) ; 0" = ~^ L ( + l )? [ " + ( /» ; $" = -^ 5>V ; K = ~ 47T E (-D q \ By usig the trasformed formulas, we get: l log <7 If log ^ ^ v \ / q - ft = J - _ 5 _ e i o g < 7 ; 3 If log q / Z Tf [- ( / ) ] ft = J--T- E^ log * ; off - T T /IZzZTWl I 7 T '\,irv/lo«q f _. ^ i 0 g ^ v log (7 ^r v log ^ r #, = ui_ CZZT E A + 7TVX ^Vlog, ^3 log q % log ^7 ~ \ log q/ ^ log (7 log ^ ^ \ log <7\ / / l ; 5, \ 7r [- ( / ) ]. COMPUTATION OF THE SUM T V T 0 L + Z We will go through the computatio of the above sum i detail. usual s 38

4 + /5 ad e = - /5 M The a3 - ad F = (a - 3 ) / V 5, L = a + If we put q = a" 5 we g e t : 0 ^ + 4 ^ + i \ ( i + q / ) By by formulas i Taery ad MoIk [7, II, pp. 50 ad 60] we have Hece, - = -TT 4- R V f \ l ( + q l ) ad i f we use formulas (3) ad ( 4 ), we g e t : E ( + l ) - [ + ( / ) ] a = I + ~ ~ [ + ( l / ) ] E a This series coverges very rapidly (0 terms will give about 0 decimal places) but it does ot cotai log a as Backstrom f s approximatio does* By usig the fuctioal equatio ad the formulas i (5) we ca improve the rate of covergece. ~ 9 9. v x /log qy ^ \ log q) s - log q E(-D ^/losq Puttig q = a, we obtai the fial formula: r L + i + 4 log a - 4TT log a E (-l) e-* / z + E ( - D e ~ i This series coverges extremely rapidly. We have e -Tr ' 8 a «e"» 0", so takig just oe term ( = ) will give over 30 correct decimal places. Te terms will give aroud 900 correct decimal places. 3. A CATALOGUE OF FORMULAS I this sectio we collect some formulas coectig sums of reciprocals of Fiboacci ad Lucas umbers ad theta fuctios. We leave it to the reader to derive the fial formulas as i the last sectio. The formulas are foud i Taery ad Molk [7, II, pp. 50, 60, 58; IV, pp. 08, 07], Jacobi [4, pp ], ad Hacock [3, p. 407]. a 986] 39

5 I. Put q = a. The: II. (b) Put (a) E : r L - L - q = a. E - - -l = ~ ( t ^ The: i) (d) t ih^-le*! I - (b) E ^ - = ^3-) (9) E T = 4 ( I + 7 T ) (d) - l ^ ^ 8TT Z #3 t t - l?^t"**^ (l)? I ^ = ^ ^ (e) E ^ F = ( ^ : - D U) E ^ 4? 4. SOME IDENTITIES There are umerous idetities amog theta fuctios. Specializig to q a" or q = a" will give idetities amog sums of Fiboacci ad Lucas umbers We will give a few examples. (a) Formulas II(i) ad (j) give two expressios for #"/# : «L. (b) Formulas (d) ad (a) g i v e, with q = a ", \ l b -\l I - \_ L h (c) The idetity (Taery ad Molk [7, II, p. 50]) o oo ^ - l co ^ ^ - 3E q - = E q - E M l -? " ) ( - t? *" ) ( l + «? - ) ( + q )' jives, with q = a" : 3 E_ E - r - = 5(z - ^ ) F 30 [Nov.

6 (d) We have $^ = # + $^9 which implies: (l + 4 f) -L-Y = M / Y + (i + 4 (-D \ \ L ll 5 \ F «- l / \ L / 5. A NEW TANTALIZING QUESTION Ufortuately, we have ot bee able to fid a expressio for the sum Sice we kow from (a) that?*, _! - 8 o g a t + t ( } & > x -l " ^ & ^ ^ we oly eed to compute ^ "TT~~* For this., we eed (wtih q = a" ) ^ I 00 ~ co a - l oo /<«> = E ^ = E -^-^-7 = E V")f 5 - q l - q _ where T 0 () is the umber of odd divisors of. Sice T Q is multiplicative 9 i.e. 5 T Q (m) = T 0 (m)t 0 () if (m 5 ) =, we ca compute the Dirichlet series (for Re s > ). oo TA) I TApv) He) = z = E = l ^ P \v>0 P vs where the product is take over all prime umbers. We have T 0 ( V ) = ad T Q (p v ) = v + if p > 3. Hece, puttig t = p~ s s we have t, o ( v ) t v = _ 4 _ ad E r o (P V >* V = E (V + l)* v = _* for p > 3. o It follows that w h e r e o H8) = l ^ = (i _ - s ) a s ) s - ~ s P>3 ( - p " s ) Us) = ±± l is the Riema ^-fuctio. It is possibles at least theoretically, to recover / from $ by Melli iversio (see Ogg [5, p..6]); however, we have ot bee able to compute the itegral. s 986] 3

7 Put We ed by givig some formulas due to Clause (see Jacobi [4 S I, p. 39]): What we eed is Mq) = E ~ 3 = E q«l±^~ - q - q -l m /. w, \ L = h(q) - h(q z ) = J f - q z - q 7 ' l \ - q - q / which coverges very rapidly whe q = a". REFERENCES. R. Backstrom. "O Reciprocal Series Related to Fiboacci Numbers With Subscripts i Arithmetic Progressio." The Fiboacci, Quarterly 9, o. (98):4-.. R. Bellma. A Brief Itroductio to Theta Fuctios. New York: Holt 5 Riehart & Wisto, H. Hacock. Lectures o the Theory of Elliptic Fuctios. New York: Dover Publicatios, C. Jacobi. Gesammelte Werke. Vol. I. Berli: Reimer A. Ogg. Modular Forms ad Dirichlet Series. New York ad Amsterdam: Bejami, H. Rademacher. Lectures o aalytic umber theory, J. Taery & J. Molk. Elemets de la theorie des fuctios elliptiques. Vols. I-IV. New York: Chelsea, [Nov.