Elementary Problems and Solutions , Edited by AP Hillman 466 Advanced Problems and Solutions ...

100 Pages·2010·19.67 MB·English

by  Karl Dilcher

THE OFFICIAL JOURNAL OF THE FIBONACCI ASSOCIATION
VOLUME 19 DECEMBER
NUMBER 5 1981
CONTENTS
A Generalized Extension of Some Fibonacci-Lucas Identities
to Primitive Unit Identities Gregory Wulczyn 385
A Formula for Tribonacci Numbers Carl P. McCarty 391
Polynomials Associated With Gegenbauer
Polynomials „ A. F. Horadam & S. Pethe 393
Enumeration of Permutations by Sequences—II L. Carlitz 398
How to Find the "Golden Number"
Without Really Trying Roger Fischler 406
Extended Binet Forms for Generalized Quaternions
of Higher Order . A. L. lakin 410
A Complete Characterization of the Decimal Fractions That Can
Be Represented as X10~k(i+1) Fai Where p Is the aith.
9 ai
Fibonacci Number ........... Richard H. Hudson & C. F. Winans 414
On Some Extensions of the Meixner-Weisner
Generating Functions M. E. Cohen & H. S. Sun 422
Almost Arithmetic Sequences
and Complementary Systems Clark Kimberling 426
Sums of the Inverses of Binomial Coefficients .. Andrew M. Rockett 433
Tiling the Plane with Incongruent Regular Polygons .. Hans Herda 437
A New Definition of Division in Rings of Quotients
of Euclidean Rings M. W. Bunder 440
A Recursion-Type Formula for Some Partitions Amin A. Muwafi 447
Primitive Pythagorean Triples and the Infinitude
of Primes Delano P. Wegener 449
An Application of Pell's Equation Delano P. Wegener 450
Central Factorial Numbers and
Related Expansions Ch. A. Charalamhides 451
On the Fibonacci Numbers Minus One G. Geldenhuys 456
Pascal's Triangle Modulo p Calvin T. Long 458
On the Number of Fibonacci Partitions of a Set .. Helmut Prodinger 463
Elementary Problems and Solutions , Edited by A. P. Hillman 466
Advanced Problems and Solutions .. Edited by Raymond E. Whitney 470
Volume Index 477
*Bfe Fibonacci Quarterly
Founded in 1963 by Verner E. Hoggatt, Jr. (1921-1980),
Br. Alfred Brousseau, and I.D. Ruggles
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A GENERALIZED EXTENSION OF SOME FIBONACCI-LUCAS IDENTITIES
TO PRIMITIVE UNIT IDENTITIES
GREGORY WULCZYN
Bucknell University, Lewisburg, PA 17837
This paper originated from an attempt to extend many of the elementary Fibo-
nacci-Lucas identities, whose subscripts had a common odd or even difference to,
first other Type I real quadratic fields and, then, to the other three types of
9
real quadratic field fundamental units. For example, the Edouard Lucas identity
^3 -, + &!! - Fl i - F„ becomes, in the Type I real quadratic field,
JL
[/Si- a = 39 y ^ 3 + 3 _ ps^
F + i 39F m (5)(195)F3ne
This suggests the Type I extension identity F„+i + L ^ - Fn-i = FFF and the
1 2 3n
Type I generalization: F% r+i .+ ^ir+i^n - Fn-ir-i = F + F i^in • Tne Ezekiel
+2 2r 1 kr+
Ginsburg identity F„ - 3F^ + Fn-2 - 3F becomes, in the Type I real quadratic
+2 3n
field,
(/61)F 3 - 1523F3 + F3. = (195) (296985)F .
n +2 n 2 3n
This suggests the Type I identity extension F„+2 ~ LiF\ + Fn-i = F Fi*Fs and the
2 n
Type I generalization: F3 r - L Fn + ^n-2r = F FkrF$ .
+2 2r 2r n
The transformation from these Type I identities to Type III identities can be
represented as
(I) F *-+ (III) 2F or (I) L ++ (III) 2L .
n n n n
The transformation from Type I to Type II and Type III to Type IV for identities
in which there is a common even subscript difference 2v can be represented as
(I, III) F +-> (II, IV) F L «-* L , F * *+ Fn+r* and L «-• L .
2r r9 2r r n+2 n+2p n+r
I. Type, I primitive units are given by
a = 2—"9 ^ = 2 s a^ = ' " (modul° 8)*
a2 - &2£> = -4, a and 2? are odd.
{s + >£f. iii^f. . i ,. - "), „...-• r.
P< (c 6
F and L are also given by the finite difference sequences:
n M
Fn + 2 - ^n l + *». *1 = *>» F = a&;
+ 2
L = aL + Zr, L = a, L = a + 2.
n+2 n+1 n x 2
11 • Type. II primitive units are given by
. SL±£& p _ i l M , - l, = 5 (modulo 8),
a t a3 D
2 _ fc2 = 4 2 _ fo2 ^ _^ £ are odd.
a D s a D a and
(iL±f^J - — -, F = ^(a» - 3n) L = a* +
2 n 9 n
F„ and £ are also given by the finite difference sequences:
n
385
386 A GENERALIZED EXTENSION OF SOME FIBONACCI-LUCAS IDENTITIES [Dec.
Fn + 2 = ^n + 1 - Fn> F! = *>9 F, = ab;
Ln + 2 = aLn+l ~ Ln> L± = a9 L2 = a - 2.
111 • T{/pe III primitive units are given by
a = a + £/D B = a - &/D, ag = -1, a2 - b/D = -1.
9
(a + b/Df - L + F v^, F = ~{an - 3n), L = |(a» + Bn) -
n n n n
F and L are also given by the finite difference sequences:
n n
F = 2aF + F, F = b F = lab;
n+2 n+1 n ± 9 2
L = 2aL + L L = a L = 2a2 + 1.
n+2 n+1 n9 ± 9 2
^' T^/pe 11/ primitive units are given by
a = a •+ 2?/D, 3 = a - &/D, ag = 1, a2 - b2D = 1, a2 - b2D £ -1.
(a + 2v£)n = L + F/D\ F = ~~(an - 3n), L„ = y(an•+ 6n).
n n n
F and L are also given by the finite difference sequences:
n n
Fn+2 = 2*Fn+l ~ ^ , ^ = £, ^ = 2aZ? 5
2
£ = 2oL - L L = a L = 2a2 - 1.
n+2 n+1 n9 1 9 2
1. (a) Fibonacci-Lucas identity used: F + L - 2F
n n
(b) Type I extension: aF + bL = 2F
n n n+1
(c) Generalizations:
Types I & II L F + F L = 2F
m n w n m + n
Types III & IV LF.+ F^L = F
n n n w+n
2. (a) Fibonacci-Lucas identity used: L - F = 2F_
n n n
(b) Type I extension: bL - aF = 2F _
n n n x
(c) Generalizations:
Type I *^B - LmFrt = 2(-ir+x.,
Type II F L - FmLn = Wn-m
*- num
Type III FmK - LmFn = V x/ L n-m
Type IV FnLm - FmLn = Fn -m
3. (a) Fibonacci-Lucas identity used: F+s + F%- = 2(F2+2 + Fn + i)
n
(b) Type I extension: b(F2 + F2) = F(F2 + F2 )
n + 3 3 n + 2 + 1
(c) Generalizations:
Types I & III F _(F ^ _ + F) = F _ (F _ + F _)
2r 1! n+ m 1 n hm 1 n + 2m + r 1 n+2m r
F Ir-l^n + km-l + ^n ) = F hm-l^n+2m + v-1 + ^ + 2^-^
Types II & IV F _ (F* _ - F2) = F ^ . ^ F ^ ^ ^ - F2 _)
2r x +hm 1 n n+2m r
"Zr-l^n+'-tm-l "" ^ n> ~ Fh m- 1 ^n + 2 m + r- 1 ~ Ln + 2m_v)
1981] A GENERALIZED EXTENSION OF SOME FIBONACCI-LUCAS IDENTITIES 387
4. (a) Fibonacci-Lucas identity used:
F ~F , + F F = 2(F F 4- F F ^
Ln + 3J-n+k T rnrn+l ^,vcn+2rn+3 T n + lr n + 2'
(b) Type I extension:
^^n+3Fn+k + ™n™n + l' = ^3 ^ n + 2^n + 3 + ^n+l^n + 2^
(c) Generalizations:
Types I & III
•^ 2r- 1 ^ n+ km-1 n+ km n*n+l' ~ km- 1 ^ n + 2rn + r- 1 ^n+2m+ r n + 2m-Kn+2w-r+P
2r-l' n + ta-1 n + ^m ttn+l' = km-1^ n + 2m + r>-1 n+2m+r n + 2m- r^n+ 2m- r + 1'
Types II & IV
2r-l^ n+km-1 n+km n n+1' km-1^ n+2m+r-l n+2m+r n+2m-r n+2m-r+1^
F2r-l^n+km-l^n+km ~ ^n^n+l' = Fkm- 1 ^n + 2m + r- l^n + 2m+ r " n+ 2m-r^n + 2m- P+ 1'
5. (a) Fibonacci-Lucas identity used: F + F% = 2F F
2m m m+1
(b) Type I extension: £F + aF* = 2i^F
2m m w+1
(c) Generalizations:
Type I FF + LF* = 2F F
r 2m r m m+r
DFF + LLl = 2L L
r 2m v m m+r
T7Pe II ^ +LFl = 2F F
2 r a v m m + p
D ^ + LL2 = 2£ L
2 m r n m m+r
lyPe " I FF + 2L F* = 2F F
r la P m m+r
DFrF2a + 2 L ^ - = 2 L ^ m + r
Type IV FF + 2LF2 = 2F F
r 2m r m m m+r
DFrF + 1L*Ll = lLmL^r
2m
6. (a) Fibonacci-Lucas identity used: F ~~ ^m ~ 2-FmFm-i
2m
(b) Type I extension: bF\ - aF* = 2F F _
m W W 1
(c) Generalizations:
Type I FF - LFl = 2{-XY^F F _
r 2m r m m r
^FrF2 ~ LL2 = ~2L L _
m r m m m r
Type III FF - 2LFl = 2(-l)*+1F F _
r 2m r m m T
DFF - 2LLl = 2{-lV+1L L _
r 2m r m m r
Type IV FF - !L Fl = -2F F _
r 2m T m m r
DFF - 2LL2 = ~2L L .
r 2m v m m mr
7. (a) Fibonacci-Lucas identity used: L\ - F% = 4F _ F
n 1 n+1
(b) Type I extension: b2L* - a2F% = 4F _ F
n 1 n+1
388 A GENERALIZED EXTENSION OF SOME FIBONACCI-LUCAS IDENTITIES [Dec.
(c) Generalizations:
Types I & III
Fill - LlFl = H-iy+1F rF -r> I; <-l'>'+13, ,*„-,. I "
n + n +
^FlFl - LlLl - ^("Dr+1L V I; <-l)1,+1L ,L _ ., III
n + r r 9 n+ ll 1
Types II & IV
FF - LF = ~bF F_, II; -^ ^ , .f IV
r n r n n+r n r n+ n 1
8. (a) Fibonacci-Lucas identity used: ^2n^2n+2 " $F2n+i ~ 1
(b) Type I extension: L L - #F| = ^2
2n 2n+2 n+1
(c) Generalizations:
AH Types L L - DF2 = L*
2n 2n+2r 2n + r
^2n+r ~ DF2nF2n+2r = ^*
9. (a) Fibonacci-Lucas identity used:
p = p p p +FFF-F^F^F^
(b) Type I extension:
GO rr+m+n = J. m + i^ n + i^ r+i + ^m^n^r "" ^ m-1^ n-l^r-1
(c) Generalizations:
TyPe 1 Fm+2t+lFn+2t+lFr+2t+ 1 + ^2i+A n *» " Fm-2t-lFn-2t-1Fr-2t-1
**J)&et+3 + ^t+l'^m + n+r = ^2£ + l^\t + 2^m + n + r
•^m+2t+l^n + 2t+l^r+2t-H ^ ^2t+l^m^n^r "~ ^m- 2t- l^n-2t- l^r- 2t- 1
~ (L$t+3 + £J2t+l)£'m+n+r = &F2t+lFkt+ 2Fm+n+ r
Fm+2tFn + 2tFr+2t ~ L2tFmFnFr + Fm-2tFn-2tFr-2t = p(^6t ~ ^2* )^m + « + r
^m+2t^n + 2t^r+2t " ^It^m^n^r + ^m-2t^n-2t^r-2t = (-^6t ~ ^2* )£/*+«+r
= ^F2tFm^m+n+r
^m + t^n + t^r+t " LtLmLnLr + Lm-tLn-tLr-t = ^3* " ^t^w + n + r
Type III ^m+2*+l n+2*+l^r+2*+l + ^2t+l"m n ^ " *w- 2t- l*n- 2£- l*r- 2t- 1
= "2D 6t+3 + ^2t+l'^m+n+r ~ ^2t + l^tH- 2^m+n+ r
^m+2t+l^n+2t+l^r+2t+l + '2t+l^m^n^r ~ "m-2t-l^n-2t-l"*-2t-1
= "2"(^6t+3 + -^2t+l)^/w + n+r = ^F2t+ lFht+ 2^m + n + r
Fm+2tFn + ZtFr+2t ~~ 2L2tFmFnFr + Fm-2tFn-2tFr- 2t = 2£^6t " L2t>>Fm + n+r
~ F2t Fht Fm + n+r
^m + 2t^n + 2t^r+2t " 2L2tLmLnLr + Lm^ 2tLn_ 2t Lr_ 2t = "J^6t " ^2t)Fm + n+r
1981] A GENERALIZED EXTENSION OF SOME FIBONACCI-LUCAS IDENTITIES 389
II2U1 Fm+tFn+tFr+t - ^AFnFr + ^ n - t V * " ^ * ~ hK+n + r
Lm + tLn + tLr+t " 2LtLmLnLr + Lm-tLn-tLv-t " ~2^L 3t " ^t^/n+n+r
10. (a) Fibonacci-Lucas identity used;
pi F2 + Fz = 2(F2 - F F )
+
(b) Type I extension:
(c) Generalizations;
TyPe I Fn+2r+l + L2r+lFn + Fn-2r-l = 2(^n + 2r+l ~ ^2r+l/n-2r-lFn•)
^n + 2r+l + ^2r+l^n * ^n-2r-l = 2(Ln + 2r+l "" ^2r + l^n- 2r- l^n)
F2 + L2 F2 + F2 = 2(F2 + L F F )
n + 2r T u2v n T £n-2v ^^Ln+2r ^ u2vL n- 2rL n '
Ln+2r + ^22>^n + ^n-2r = 2^n + 2r + L2rLn- 2rLn)
Type II F2+r + £2F2 + F2_r = 2(F2+„ + LrFnFn„r)
L\+v + LrLn + Ln-r = 2^n+r + LrLnLn-r)
Type III ^n+2r+l + ^Zr+lFn + ^n-2r-l = 2Wn+2r+l " 2L2r+ l^n-2r- l^n)
^n+2r+l + **L2r+iLn + £n-2r-l = 2(-^n+2r+l ~ 2Zr2r+ i£n- 2r- l^n)
F + kL F + F _ = 2(F + 2L F_ F)
n+2r 2v n n 2r n+2r 2r n 2r n
£Jn + 2r + **L L + L_ = 2(L , + 2L L _ ,L )
2r n n 2r n + 2r 2r n 22 n
Type IV F2 + 4L2F2 + ^ - r = 2(F2 + 2L F F _ )
+r +r r n n r
L + 4L L + L _ , = 2(L + 2LLL_)
n+r r n n 2 n + r r n n r
11. (a) Fibonacci-Lucas identity used;
Fn+2 = ^n + ^n+1 + 3FnFn+lFn+2
(b) Type I extension;
P3 = K+ ^FUi + 3oF„F F
n + 2 n+1 n+2
(c) Generalizations:
Type I Fl + l = F*n-2r-l + ^lUl^ + 3L2r l^ ^n + 2r + l^»-2p-l
2r+ +
•^n + 2r+l = ^n-2r-l "*" ^2r+l^n + 3L L £ +2r+l^n-2r-1
2r+1 n
F = L F - F . - 3L F _ F F„+2t
n+2t 2t n n2t 2t n 2t n
^n + 2£ = ^2t^n ~ ^n-2t ~ 3L2tLn- 2t^n^n+2t
Type II F 3 , = L3F 3 - F3_ - 3L F F _„F
n +2 n n r r n n M+r
^n+r = LrLn ~ Ln-r *" ^LrLnLn-rLn + r
Type III ^ = ^-2,-1 + 8LL iFn + 6L F F F . .
+2r+1 + 2p+1 n n + 2r+1 n 2r 1
L+2r+l ~ £n-2r-l + 8L L + 6L Iy L , L _ _
n 2r+1 n 2p+1 n n + 22 + 1 n 2r x
^n+2t = &L F - F _ - 6L F_ FF
2t n n 2t 2t n 2t n n + 2t
Ln + 2t = 8^2t:^n ~Ln-2t " ^L2tLn^2tLnLn + 2t
390 A GENERALIZED EXTENSION OF SOME FIBONACCI-LUCAS IDENTITIES [Dec.
Tvr>e IV F3 = SL3F3 - F3 - 6L F F F
£3 = 8LrLl " L\~v " 6LrLnLn-rLn+r
n + r
12. (a) Fibonacci-Lucas identity used:
K l + Fn + *•»-! = 2[F„2 - F ^ ^ ] 2
+ +1
(b) Type I extension:
(c) Generalizations:
2
TyPe T ^n+2r+l + ^2r+A + ^n-2r-l = 2lFn+2r+l ~ ^2r+ l^n "^'n -2r- 1 J
•^n + 2p+l + ^2p+l^n + ^n-Zr-1 = 2l^n+Zr+l ~ ^2r+ l^n^n - 2v- 1 J
rc + 2£ T ^2* n Trn-2£ zLrn+2t T Lj2trnr n - It J
^n+2t + ^2t^n + ^n-Zt = 2[^n + 2t + ^2t^n^n-2t]
Type II # + L ^ + F4_ = 2[F 2 + L ^ ^ . J 2
+P n p n +r
^n + r + ^p^ + -^n-p = 2^n+r + ^r^n^n-ri
Type III £7+22>+l + 16i2r+l^n + Fn-2r-l = 2[^n + 2r+l ~ 2^2P+l^n ^n - 2r - 1 ]
n
^rc+2p+l + 16^2r+l"C'n + ^n-2r-l = 2^-Ln + 2r+l " 2^2r+A^n-2r-J
^n + 2t + 16^2£Frc + ^n-2t = 2lFn+Zt + 2i2t ^n ^n-2t ]
^n+2£ + ^^2t^n + Ln-2t = 2[Ln+Zt + 2LLL_]
lt n n lt
Type IV # . + 16LX + ^-r - 2[** ..+ 2L F F „ ]2
+2 + 2 p n n r
^n+P + 16LrLl + £ L P = 2^-Ll+r + 2LrLnLn-r^2
13. (a) Fibonacci-Lucas identity used:
Fn+1 ~ Fn ~~ Fn-1 = 5^n Fn- lFn +1 (^n+1 ~ Fn-lFn)
(b) Type I extension:
Fl + l ~ & Fn ™ Fn-1 = 5aFnFn-lFn + l(Fn+l " ^n-l^n)
(c) Generalizations:
Type I
•^n+2r+l ~ ^2r+l^n " Fn-Zr-1 = ^ p+A^n~2r-A+2r+l ft+2p+l " ^Zr + lFFn - Zv - 1)
2 n
•^n + 2r+l " -^2r+l^n " ^n-Zr-1 = ^2r+ l&n^n- Zv - l^n + 2r+l (^n+ 2r + 1 "" ^2r+ lLL _ 2 l)
n n
^ZtFn " ^n+2t ~ Fn-Zt = ^ 2tFF n-2tFn+Zt (Fn + Zt + ^2tFnF n-2t)
n
^2t^n ~ ^n+2t ~ ^n~2t ~ ^zt^n^n-2t^n + 2t (^n + Zt + ^Zt^n^ n-Zt)
Type II L5F5 - F 5 - ^ _ LF F _ F (F2 + L F,F _ )
r n n +p r a 5 r n n n + JJ +p r n p
^r^n ~ Ln+r ~ ^ n-r = ^r^n^n- ^n+r(Ln+r + ^^n^ n- r) '
r
Type III
7T5 _ 32T/5 7^5 - F 5 = 10T/ F F 7J7 CF2 - IT, F F }
Ln+2r+l ^^Zr+Y- n Ln-2v-\ xyj-u2r+ \L n n- 2v- 1L n+ 2r+ 1 vL n+ 2r+ 1 z'1J2r+ 1 n n- 2r- 1'
^n + Zr+1 " ^2"^2p+l^n "" ^n-2r-l ~ ^^2r+ l^n^n - Zv - l^n + Zr+ 1 ^n + 2r+ 1 "" 2j^2r+ l^n^n - Zr- 1'
1981] A FORMULA FOR TRIBONACCI NUMBERS 391
32L5 F5 - F5 - F5 = IOL F F F (F2 + IT, F F *\
2t n rn + 2t £ n-lt LKJLj2trnrn- 2tr n + 2t ^n + 2t + LLj 2t n n - it '
32L2tLn - Ln + 2t ~ Ln-2t ~ lQ£/2tLn£Jn-2t£'n + 2t (^n+2t + 2^2t LnLn _lt)
Type IV
32L5F* - F5 - F5 _ = lOLFF_F (F* + 2LFF_)
r n + T n r r n n r n + r +r r n n r
^2LrLn " Ln+r> " Ln-r = l0LrLnLn- rLn + r(L n+ r + 2Lr^nLn_r)
14. (a) Fibonacci-Lucas identity used: L\ = 2Fn-i + F\ + 6Fn+i^ -i
n
(b) Type I extension: £>3L3 = 2F3_ + a3F3 + 6F2 F _
1 i + 1 n 1
(c) Generalizations:
TyPe I F2r+lLn = 2Fn-2r-l + LZr+lFn + 6^n + 2r+1F n-2v- 1
•^ ^Zr+l^n = 2Ln_2r-l + ^2r+l^n + 6Ln+2r+ l^n- 2r- 1
F2r^n = £J2rFn ~ 2Fn-2r ~ ^n+2r^n-2r
D FZrFn = L2r£Jn ~" 2^n-2r ~ 6Ln + 2r^n - 2r
TvDe II F3L = L3F - 2F - 6F F
D FF = L L - 2£ _ - 6^ + pLn-p
r n p n n r n
Type III 4F3 L3 = F3_ ,-i + ^3 F3 + 3 ^ F . _
r+1 2 2r+1 + 2r+1 n 2r x
A/?73 r3 = 4r,3 F3 - F3 - ^p2 /?
ti. 2r±J n ^±J2vJ-n •Ln-2r ~>I- n+ 2V1- n - Zv
4£ F2rFn = ^2r^n ~ ^n-Zr " 3Ln+22,Ln_2p
Type IV 4F3L3 = 4L3F3 - F3_ - 3F* F _
r n r +r n r
kD FF = 4L L - L _ - 3L ,L„_
v n p n n r n +2 r
Concluding Rma/ilu
Following the suggestions of the referee and the editor, the proofs of the 14
identity sets have been omitted. They are tedious and do involve complicated, al-
beit fairly elementary, calculations. For some readers, the proofs would involve
the use of composition algebras which are not developed in the article and which
may not be well known.
The author has completed a supplementary paper giving, with indicated proof,
the Type I, Type II, Type III, and Type IV composition algebras. After each com-
position albegra the corresponding identities using that algebra have been stated
and proved. Copies of this paper may be obtained by request from the author.
A FORMULA FOR TRIBONACCI NUMBERS
CARL P. MCCARTY
LaSalle College, Philadelphia, PA 19141
In a recent paper [2], Scott, Delaney, and Hoggatt discussed the Tribonacci
numbers T defined by
n
T = 1, T = 1, T = 2 and T = T _ + T-i + T _ , for n >_ 3,
Q ± 2 n n x n n 3
392 A FORMULA FOR TRIBONACCI NUMBERS [Dec.
and found its generating function, which is written here in terms of the complex
variable g, to be
(1) /(*) - — — - — - £ T z».
n
1 - z - z2 - z3 "=0
In this brief note, a formula for T is found by means of an analytic method sim-
n
ilar to that used by Hagis [1].
Observe that
(2) z3 + z2 + z - 1 = (a - r)(z - s)(z - "s),
where r - .5436890127,
8 = -.7718445064 + 1.115142580£,
\s\ = 1.356203066,
and
\r - s\ = 1.724578573;
thus f(z) is meromorphic with simple poles at the points z = r z = s and z = IF,
9 9
all of which lie within an annulus centered at the origin with inner radius of .5
and outer radius of 2.
By the Cauchy integral theorem,
r -./(n)(0) l f /<«> dz
= IU J
2TT£ # sn+i
1*1 -.5
and by the Cauchy residue theorem,
(3) ^ l i j ^ - ^ ^ ^ ^
Ivi J „-i
z
1*1-*
where R J> 2 and i?, i? and i? are the residues of f(z)/zn+1 at the poles r s
x 25 3 9 9
and s, respectively.
In particular, since f(z) = -l/((z - r)(z - s)(s - IF)),
(4) i?, = lim (a - r)f(z)/zn+1 = -l/((r - s) (r - s")rn+1)
js + r
= -l/(|r - sl2!'^1),
(5) • B = lim (z - s)f(z)/zn+1 = -l/((s - r) (a - s)sra+1),
2
and
(6) i? = lim (s - ~s)f(z)/zn+1 = -l/((s - r)(s - s)¥n+1) = i?
3 2
Along the circle \z\ = R >. 2 we have
z3 + s2 + z - ll |U|3 - Is2 + 2 - ill i?3 - R2 - i? - 1
hence
11 f/ //((ga)) dd z
(7)
2TT£ i rJ 2n + i i? (i?3 - i?2 - i? - 1)
1*1-/?
Now, if R is taken arbitrarily large, then from (3) and (7.) it follows that
(8) T = -(R + R + i? ).
n 1 2 3
One final estimate is needed to obtain the desired formula. From (5) we have
for n > 0,