SUMMATION OF RECIPROCAL SERIES OF NUMERICAL FUNCTIONS
OF SECOND ORDER
BLAGOJ S. POPOV
"Kiril
i Metodij/'
Skopje,
(Submitted
September
1983)
University
Yugoslavia
This paper is an extension of the results of G. E. Bergum and V. E. Hoggatt9
Jr. [1] concerning the problem of summation of reciprocals of products of
Fibonacci and Lucas polynomials™ The method used here will also allow us to
generalize some formulas of R. Backstrom [2] related to sums of reciprocal
series of Fibonacci and Lucas numbers.
1
The general numerical functions of second order which, following the notation
of Horadam [3]9 we write as {wn(a5 b; p s q)} may be defined by
wn = pwn_±
- qwn_2$
n > 2 9 w0 = a9 w1 = b5
wn = wn(a9
b; p, q)
with
s
where a and b are arbitrary integers,
We are interested in the sequences
un = wn(0s
1; p 9 q)
(!)
vn = wn(29
p; p 5 q)
(2)
and
that can be expressed in the form
u
n
n
= ^ ^ ,
a - 3
»>1,
(3)
n > 1,
(4)
and
vn = an + 3n>
where
a = (p + Vp2 - 4q)/2s g = (p - Vp2 - 4^)/29 a + 3 = P* ag = qs
and
a - 3 = 6 = AUsing (3) and (4) 9 we obtain
2a n = vn + 6u n
and
4am+n =
^ ^
+ t\umUn
+ 6(umyn +
UmVn)s
from which it follows that
U
1 986]
8 + PVB
~
U V
s s
+v
=
2c Su
? r>
( 5 )
17
SUMMATION OF RECIPROCAL SERIES
and
V
s+rVs
~ &UsUs
(6)
2q8VP.
=
+ r
From r e l a t i o n ( 5 ) , we have
v,
vaA
Ua
Ua
= 2q*
If we replace s here by s, s + P, s + 2P, . .., s + (n - 1)P, successively,
and add the results, we obtain, due to the telescoping effect,
s n (p, q; p, s ) = E
fc=l
M
- — = I— - - — ) — s —
s + (/<-l)rus+fcr
\ws
u
s+nrl2qbUr
UrU sUs
+
(7)
np
Similarly, again using (5), we also have
On(p9
q; P , s ) =
E
k=l
V^ us+kr
{k-l)r
u t
V
s +
^ / 2 2q
< 7suS T
~ U s + wr
urVsVs+?
(8)
Because
"P,
|B/a| < 1
lim
n + °° U-n + I
and
| a / g | < 1,
fa^/Ca2
lim
- q)9
U'-^/CB2 - q),
|g/a| < 1
|a/B| < 1,
we o b t a i n
5 ( p , <?; P , s ) =
£
7(k-
-,
l)p
(9)
/ c = l Ws + ( / c « i ) r W s + fo
M
rV
a1"3
r ( p , q; r,
s) =
|3/a| < 1
£
7 (fc-
2
1
D^
fc = l ^ s + C f e - D r ^ s + fcr
a
- q
|U,/
1
,
upys
|B/a| < 1
(10)
gl-e
e2 - < ?
p
1
W
^S
,
|a/B| < 1.
In particular, with r = s, we have
5(p, 4;
18
P,
p)
(11)
[Feb.
SUMMATION OF RECIPROCAL SERIES
and
- q2r)s
a'/Ja^
0(p9
q; r ,
|B/a| < 1
r)
(12)
&*/&** - q2r)>
3.
| a / g | < 1.
SPECIAL CASES
It is not difficult to obtain the formulas of Bergum and Hoggatt from (9) and
(10). Indeed, if we let p = x and q - -1 in (1) and (2), these relations define the sequences of the Fibonacci polynomials {Fk(x)}™
and the Lucas polynomials {Lk(x)}k=1,
In this case,
i(x)
= (x + Vx2 + 4)/2,
6(a?) = (x - V i 2 + 4)/2,
where
-1 < a (a?) < 1
and
@(#) > 1
when a? > 0,
0 < a (a:) < 1
and
&(x) < 1
when x < 0.
Hences (9) and (10) become
r
£(#, - 1 ; r9
s) = lim Sn(xs
- 1 ; r 9 s) =
__l
1
•,
x > 09
1
1
•9
g8(a?) F p (x)F s (x)
x < 09
lor
as(x)
Fv(x)Fs{x)
(13)
and
a 1 " 8 (a;)
-,
1 + a 2 Or) Fp(x)Ls(a?)
a(x 9 - 1 ; r, s) = lim a„0c, - 1 ; rs
x > 0
(14)
s) =
1 + 3z(x)
,
x < 0.
Fr(x)LAx)
Comparing the results of Bergum and Hoggatt [1, p. 149, formulas (9) and
(17)] with our (13) and (14) above, we find that
U(q9
a, b, x)
= {-l)hFk{x)Fq
(x)S(xs
V(qs
a, b9 x)
= (-l)bFk(x)Fq
(x) (x2 + 4)aGc,.-1; q9 i ) ,
(15)
-1; q9 b)
and
(16)
when q = b ~ a + k.
As particular cases, we give:
f 2
S(x.
- 1 ; 2, 2) =
E
^
1
^
$ (x)/x25
^
^
a2fe)/x2,
x > 09
aj < 0,
and
a2(x)/(a8(x)
crfe, - 1 ; 2, 2) = £
1986]
-r-y-
r-y
- 1),
J: > 0,
B 2 te)/(e 8 (*) - 1), x < 0.
19
SUMMATION OF RECIPROCAL SERIES
(5) and (6) with u_n = -q~nun
Using the r e l a t i o n s
Then, by the method used to obtain
jkr
U
(2k-h l ) r + 2.3
U
^
we have
( 7 ) , we have
U
1
A£
and v_n = q^nvn,
(n+l)r
(17)
r
so t h a t
7s
-,
+ kr
|3/ot| < 1,
(18)
AE
k = 0 7,
_ ^ s + fcr7,
-,
Iot/3 I < I -
Similarly, from
y 2 r + (? p " s y 25
= vr_svr
+i
using (8) we obtain
u
1
(19)
s +1
/(<7 - 3 r ) w r y s ,
rs + kr
k=0
V(2k+l)r+2s
(n+l)r
+ kr^V
+ <l
[us+1/(q
v
| e / a | < 1,
" ar)urvsi
(20)
| a / 6 | < 1.
In particular, if we put p = -q = 1 in (17)-(20), we obtain the formulas
of Backstrom [2] concerning the Lucas numbers. These are
1
£
(n+l)r
*>F F F
k = 0^(2k+
l)r+ 2s + ^ r
,
1
F(n + l)r
FrLg £(„ + 1 ) 2 , + i ,
s odd,
s even,
and
i—y—jiiTv
fe = o £ ( 2 f c + l ) r + 2 s
+
L
v's - l \ s
r
( ^ )
1
-,
sodd
-
s even,
where p is an even integer satisfying -r ^ 2s ^ r - 2.
We notice that, from
w* - ^ - ^ = uv_sur
+s
it follows that
20
[Feb.
SUMMATION OF RECIPROCAL SERIES
v
±
Sn(ps
n
{2k - l)r+ s
q; 2PS s)
(21)
v
and
Q2(n-l)r
V
-
k=l
u
U
2
(2k-
-
l)r+s
n
s +
4
2 A : PU
7,2
v
($s/u2rus*
| 3/ot| < 1,
[a s /w 2p w sS
|a/g| < 1.
Similarly 9
n
02(k-l)r
A \ "
k =l
2
. ..
„s+2kr, ,2
(2fc- 1 ) P + S
<7
7y )
= Sn(p5
q; 2r9
s).
ACKNOWLEDGMENT
The author wishes to thank the referee for his comments and suggestions which
have improved the presentation of this paper.
REFERENCES
1.
2.
3.
4.
5.
G. E. Bergum & V. E. Hoggatt9 Jr. "Infinite Series with Fibonacci and
Lucas Polynomials.11 The Fibonacci
Quarterly
17, no. 2 (1970):147-151.
R. P. Backstrom. "On Reciprocal Series Related to Fibonacci Numbers with
Subscripts in Arithmetic Progression." The Fibonacci
Quarterly
19, no. 1
(1981):14-21.
A. F. Horadam. "Generating Functions for Powers of Certain Generalised
Sequences of Numbers." Duke Math, J, 32, no. 3 (195.6) :437-446..
A. F. Horadam. "A Generalised Fibonacci Sequence." Amer. Math,
Monthly
68 5 no. 5 (1961)^455-459.
A. F. Horadam. "Basic Properties of a Certain Generalized Sequence of
Numbers." The Fibonacci
Quarterly
3 (Oct. 1965):161-177.
•0404
1986]
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