Closed Form Evaluation Of Melham's Reciprocal Sums

Vol. 18 (2017), No. 1, pp. 251–264 DOI: 10.18514/MMN.2017.1284

CLOSED FORM EVALUATION OF MELHAM’S RECIPROCAL SUMS

E. KILIC¸ AND H. PRODINGER

Received 13 June, 2014

Abstract. Recently Melham [1] gives closed formulæ for certain finite reciprocal sums. In this paper, we present a different approach to compute these sums in closed form. Our approach is straight-forward and simple. First we convert the sums in q-notation, then use partial fraction decomposition and telescoping to derive closed formulæ.

2010 Mathematics Subject Classification: 11B39

Keywords: reciprocal sums identities, partial fraction decomposition

1. INTRODUCTION

Let a 0 and b  0 be integers with .a; b/ ¤ .0; 0/. For p a positive integer, define the sequencesfWng and fWng by

WnD pWn 1C Wn 2 and WnD Wn 1C WnC1;

where W0D a, W1D b.

When .a; b; p/_{D .0; 1; 1/, we have fW}ng D fFng, and fWng D fLng, which are

the Fibonacci and Lucas numbers, respectively. When .a; b; p/D .0; 1; p/, we denote fWng D fUng, and fWng D fVng, which are the generalized Fibonacci and Lucas

numbers, respectively.

For k 1, m  0, and n  2, the author [1] gave closed formulæ for the following finite reciprocal sums:

(1) For nonnegative integers m1< m2and m3< m4with m1C m2D m3C m4, n

X

t D1

1

W_{k.t Cm1/Cm}W_{k.t Cm2/Cm}W_{k.t Cm3/Cm}W_{k.t Cm4/Cm};

(2) For nonnegative integers mi with mi < mi C1 for 1 < i < 4 and positive

integer c, i) n X t D1 U_{2k t C2m} P6.W; W / ii) n X t D1 U_{2k.t Cm1/C2m} P6.W; W / iii) n X t D1 U_{2k.t Cm2/C2m} P6.W; W / ; c

(3) i) n X t D1 1 P8.W; W / ii) n X t D1 U_{2k.t Cc/C2m}U_{2k.t C2c/C2m} P8.W; W ; c; 2c; 3c/ ; (4) i) n X t D1 U_{2k.t Cm1/C2m}U_{2k.t Cm2/C2m}U_{2k.t Cm3/C2m} P10.W; W / ii) n X t D1 1 P12.W; W / ; where P2r.W; W ; m1; m2; : : : ; mr 1/, or briefly P2r.W; W /, is given by

P2r.W; W /D W_{k t Cm}W_{k t Cm} r 1

Y

i D1

W_{k.t Cmi/Cm}W_{k.t Cmi/Cm}:

In this paper, we present a different approach to compute these sums. Our approach is straight-forward and simple and works in all instances. First we convert these sums in q-notation and use partial fraction decomposition. Using telescoping, we derive closed formulæ for them.

The Binet forms are WnD

A˛n Bˇn

˛ ˇ D A˛

n 1.1 qnB=A/

.1 q/ , WnD A˛nC BˇnD A˛n.1C qnB=A/

and UnD ˛n ˇn ˛ ˇ D ˛ n 1.1 qn/ .1 q/ ; VnD ˛ n C ˇnD ˛n.1C qn/;

where ˛, ˇ_{D .p }pp2_{C 4 /=2, q D ˇ=˛, ˛ D iq} 12, AD b aˇ and B D b a˛. We frequently denote the sequencesfWng and fWng by fWn.A; B/g and

fWn.A; B/g, respectively.

2. SIMPLE EVALUATION OFMELHAM’S SUMS

1. We start with the first kind of sums by converting them into q-notation:

n X t D1 1 W_{k.t Cm1/Cm}W_{k.t Cm2/Cm}W_{k.t Cm3/Cm}W_{k.t Cm4/Cm} D .1 q/ 2 A4_˛k.m1Cm2Cm3Cm4/C4m 2  n X t D1 q2k t .1 qk.t Cm1/Cm_{B=A/.1 q}k.t Cm2/Cm_B=A/  1 .1_{C q}k.t Cm3/Cm_{B=A/.1C q}k.t Cm4/Cm_B=A/  ;

where m1, m2, m3and m4are defined as before.

Without constant factor and writing ´D qt kand qkmiCm_{D q}ci _{and B=AD a for} 1 i  4, we define SnWD n X t D1 ´2

.1 a´qc1_{/.1 a´q}c2_/.1C a´qc3_/.1C a´qc4_/: Let

T .´/_WD ´

2

.1 a´qc1/.1 a´qc2/.1C a´qc3/.1C a´qc4/: The partial fraction decomposition of T .´/ is

´2

.1 a´qc1_{/.1 a´q}c2_{/.1C a´q}c3_{/.1C a´q}c4_/

D q c4 a2_.qc2_{C q}c4_/.qc1_{C q}c4_/.qc3 _qc4_{/.1C aq}c4_´/ qc2 a2_.qc2_{C q}c3_/.qc2_{C q}c4_/.qc1 _qc2_{/.1 aq}c2_´/ C q c1 a2.qc1C qc4_/.qc1C qc3_/.qc1 _qc2_{/.1 aq}c1_´/ C q c3 a2_.qc3 _qc4_/.qc2_{C q}c3_/.qc1_{C q}c3_{/.1C aq}c3_´/:

The assumption m1< m2and m3< m4 with m1C m2D m3C m4means c1< c2

and c3< c4with c1C c2D c3C c4. We write

´2

.1 a´qc1_{/.1 a´q}c2_{/.1C a´q}c3_{/.1C a´q}c4_/  a2.qc2 C qc4_/.qc1 C qc4_/.qc3 _qc4_/.qc2 C qc3_/.qc1 _qc2_/.qc1 C qc3_/ D qc3_.qc2_{C q}c4_/.qc1_{C q}c4_/.qc1 _qc2_/ h 1 1_{C aq}c4_´ 1 1_{C aq}c3_´ i qc2_.qc1_{C q}c4_/.qc3 _qc4_/.qc1_{C q}c3_/ h 1 1 aqc2_´ 1 1 aqc1_´ i and so a2qc1Cc2_.1 C qc4 c2_{/.1 q}c4 c3_/.1 C qc3 c2_{/.1 q}c2 c1_/.1 C qc3 c1_/S n D .1 C qc4 c2_{/.1 q}c2 c1_/  n X t D1 1 1_{C aq}c3Cd_´ n X t D1 1 1_{C aq}c3_´  .1 qc4 c3_/1 C qc3 c1_/  n X t D1 1 1 aqc1Cc´ n X t D1 1 1 aqc1´  ;

which, by the telescoping sum identity n X t D1  1 1_{C a´q}bCc 1 1C a´qc  D b X t D1  1 1C a´qnCc 1 1C a´qc  gives us a2qc1Cc2_{.1C q}c4 c2_{/.1 q}c4 c3_{/.1C q}c3 c2_{/.1 q}c2 c1_{/.1C q}c3 c1_/S n D .1 C qc4 c2_{/.1 q}c2 c1_/ c4 c3 X t D1  1 1C a´qnCc3 1 1C a´qc3  .1 qc4 c3_/.1 C qc3 c1_/ c2 c1 X t D1  1 1 a´qnCc1 1 1 a´qc1  : We write it in original form

.B=A/2qk.m1Cm2/C2m_{.1C q}k.m4 m2/_{/.1 q}k.m4 m3/_/  .1 C qk.m3 m2/_{/.1 q}k.m2 m1/_/.1 C qk.m3 m1/_/S n D .1 C qk.m4 m2/_{/.1 q}k.m2 m1/_/  k.m4 m3/ X t D1  1 1_{C q}t kCnCkm3Cm_B=A 1 1_{C q}t kCkm3Cm_B=A  .1 qk.m4 m3/_{/.1C q}k.m3 m1/_/  k.m2 m1/ X t D1  1 1 qt kCnCkm1Cm_B=A 1 1 qt kCkm1Cm_B=A  or SnD A3B 2˛2.2mCkm1Ckm2/
2Vk.m4 m2/Vk.m3 m1/Vk.m3 m2/Uk.m2 m1/Uk.m4 m3/  
Vk.m4 m2/Uk.m2 m1/ k.m4 m3/ X t D1 ˛t kCnCkm3Cm W_{t CnCkm3Cm} ˛t kCkm3Cm W_{t kCkm3Cm}  C Uk.m4 m3/Vk.m3 m1/ k.m2 m1/ X t D1 ˛t kCnCkm1Cm W_{t kCnCkm1Cm} ˛t kCkm1Cm W_{t kCkm1Cm}  and so n X t D1 1 W_{k.t Cm1/Cm}W_{k.t Cm2/Cm}W_{k.t Cm3/Cm}W_{k.t Cm4/Cm}

D 1 AB2_V k.m4 m2/Vk.m3 m1/Vk.m3 m2/Uk.m2 m1/Uk.m4 m3/  
Vk.m4 m2/Uk.m2 m1/ k.m4 m3/ X t D1 ˛t kCnCkm3Cm W_{t CnCkm3Cm} ˛t kCkm3Cm W_{t kCkm3Cm}  C Uk.m4 m3/Vk.m3 m1/ k.m2 m1/ X t D1 ˛t kCnCkm1Cm W_{t kCnCkm1Cm} ˛t kCkm1Cm W_{t kCkm1Cm}  ; which is the desired evaluation of the first type of sums.

For example, we have

n X t D1 1 F_{t C1}F_{t C2}L_{t C1}L_{t C2} D 1 2 p 5˛ nC2 L_nC2 ˛2 L2  C˛ nC2 F_nC2 ˛ 2  and n X t D1 1 FtFt C3Lt C1Lt C2 D 1 6  2p5˛ nC2 L_nC2 ˛2 3  X3 t D1 ˛t Cn F_{t Cn} ˛t Ft  : 2. i) The sums take in q-notation the form

n X t D1 U_{2k t C2m} P6.W; W / D A 6˛5 2km1 2km2 4m_{.1 q/}2  n X t D1 ˛ 4k t.1 q2k.t Cm2/C2m_/ .1 q2k t C2m.B=A/2/ 1 .1 q2k.t Cm1/C2m_.B=A/2_{/.1 q}2k.t Cm2/C2m_.B=A/2_/  : Without constant factor and writing ´D q2t k, q2kmi_{D q}ci_{, q}2m_{D c and .B=A/}2_{D a} for 1 i  2, we consider SnWD n X t D1 ´.1 c´/

.1 ac´/.1 ac´qc1_{/.1 ac´q}c2_/: Let

T .´/_WD ´.1 c´/

.1 ac´/.1 ac´qc1_{/.1 ac´q}c2_/: The partial fraction decomposition of T .´/ is

T .´/_D ´.1 c´/

.1 ac´/.1 ac´qc1_{/.1 ac´q}c2_/

D .1 a/

.1 aqc2_/

a2cqc1_{.1 acq}c2_{´/.1 q}c2_{/.1 q}c2 c1_/

C .1 aq

c1_/

a2_cqc1_{.1 q}c1_{/.1 acq}c1_{´/.1 q}c2 c1_/: The assumption 0 < m1< m2means c1< c2. By telescoping, we write

SnD .1 a/ a2_{c.1 q}c1_{/.1 q}c2_/ c1 X t D1  1 1 acqn_´ 1 1 ac´  .1 aqc2_/ a2cqc1_{.1 q}c2_{/.1 q}c2 c1_/ c2 c1 X t D1  1 1 ac´qnCc1 1 1 ac´qc1  and so we obtain n X t D1 U_{2k t C2m} P6.W; W / D .A 2 _B2_/ A2_B4
_U 2km1U2km2  2km1 X t D1  ˛2t kCnC2mC3 W_{2t kCnC2m}.A2_{; B}2_/ ˛2t kC2mC3 W_{2t kC2m}.A2_{; B}2_/  W2km2.A 2_{; B}2_/ A2_B4_U 2km2U2k.m2 m1/  2k.m2 m1/ X t D1  ˛2t kCnC2km1C2mC3 W_{2t kCnC2km1C2m}.A2_{; B}2_/ ˛2t kC2km1C2mC3 W_{2t kC2km1C2m}.A2_{; B}2_/  ; where
is defined as before.

ii) Now we write the second class of sums in q-notation as

n X t D1 U_{2k.t Cm1/C2m} P6.W; W / D A 6˛5 2km2 4m_{.1 q/}2  n X t D1 q2k t.1 q2k.t Cm1/C2m_/ .1 q2k t C2m_.B=A/2_{/.1 q}2k.t Cm1/C2m.B=A/2/ !  1 .1 q2k.t Cm2/C2m_.B=A/2_/  :

Without constant factor and writing ´_{D q}2t k, q2kmi _{D q}ci _{and .B=A/}2 _{D a for} 1_{ i  2, we define the sum}

SnWD n

X

t D1

´.1 c´qc1_/

Let

T .´/_WD ´.1 c´q

c1_/

.1 ac´/.1 ac´qc1_{/.1 ac´q}c2_/:

Similarly as before, by using the partial fraction decomposition of T .´/ and telescop-ing, since c2> c1, we obtain

SnD 1 a2_{c.1 q}c2 c1_/  _{1 a} 1 qc1 c1 X t D1  1 1 ac´qn 1 1 ac´  1 aqc2 c1 1 qc2 c2 X t D1  1 1 ac´qn 1 1 ac´  : Thus we get the result

n X t D1 U_{2k.t Cm1/C2m} P6.W; W / D ˛ 2mC3 B4_A2_U 2k.m2 m1/  .A 2 _B2_/
U2km1 2km1 X t D1  ˛2k t Cn W_{2t kC2mCn}.A2_{; B}2_/ ˛2k t W_{2t kC2m}.A2_{; B}2_/  W2k.m2 m1/.A 2_{; B}2_/ U2km2 2km2 X t D1  ˛2k t Cn W_{2t kC2mCn}.A2_{; B}2_/ ˛2k t W_{2t kC2m}.A2_{; B}2_/  ; where
is defined as before.

For the next sums, we only present the key steps, as the procedure is always the same.

iii) Consider the sums and its q-form

n X t D1 U_{2k.t Cm2/C2m} P6.W; W / D A 6˛5 2km1 4m_{.1 q/}2  n X t D1 q2k t.1 q2k.t Cm2/C2m_/ .1 q2k t C2m_.B=A/2_{/.1 q}2k.t Cm1/C2m.B=A/2/  1 .1 q2k.t Cm2/C2m_.B=A/2_/  :

Without constant factor and rewriting the parameters, we consider the sums SnWD

n

X

t D1

´.1 c´qc2_/

.1 ac´/.1 ac´qc1_{/.1 ac´q}c2_/: Here by the partial fraction decomposition

´.1 c´qc2_/

.1 ac´/.1 ac´qc1_{/.1 ac´q}c2_/D 1 a2c  .a qc2/ .1 ac´/.1 qc1_{/.1 q}c2_/ qc2 c1_{.1 a/} .1 acqc2_{´/.1 q}c2_{/.1 q}c2 c1_/ .a qc2 c1_/ .1 qc1_{/.1 acq}c1_{´/.1 q}c2 c1_/  ; and using telescoping, since c2> c1, we write

SnD 1 a2_{c.1 q}c2 c1_/  a.1 qc2 c1_a 1_/ 1 qc1 c1 X t D1  1 1 ac´qn 1 1 ac´  Cq c2 c1_{.1 a/} 1 qc2 c2 X t D1  1 1 ac´qn 1 1 ac´ 

and obtain the result

n X t D1 U_{2k.t Cm2/C2m} P6.W; W / D 1 B4A2U2k.m2 m1/  _W 2k.m2 m1/.B 2_{; A}2_/ U2km1 2km1 X t D1  ˛2k t C2mCnC5 W_{2k t C2mCn}.A2_{; B}2_/ ˛2k t C2mC5 W_{2k t C2m}.A2_{; B}2_/  C .A 2 _B2_/ .˛ ˇ/U2km2 2km2 X t D1  ˛2k t C2mCnC3 W_{2k t C2mCn}.A2_{; B}2_/ ˛2k t C2mC3 W_{2k t C2m}.A2_{; B}2_/  : 3. i) For 0 < m1< m2< m3, we consider the following sums in its q-form

n X t D1 1 P8.W; W / D A 8˛8 2km1 2km2 2km3 8m_{.1 q/}4  n X t D1 ´2

.1 ac´/.1 ac´qc1_{/.1 ac´q}c2_{/.1 ac´q}c3_/: Without constant factor, we only consider the sum

SnWD n

X

t D1

´2

.1 ac´/.1 ac´qc1/.1 ac´qc2/.1 ac´qc3/:

By partial fraction decomposition of the summand and using telescoping, we write a2c2qc1Cc2_{.1 q}c1_{/.1 q}c2_{/.1 q}c3_{/.1 q}c3 c2_{/.1 q}c2 c1_{/.1 q}c3 c1_/S n D qc3_{.1 q}c1Cc2 c3_{/.1 q}c3_{/.1 q}c2 c1_/ c2 X t D1  1 1 acqn_´ 1 1 ac´  qc3_{.1 q}c2 c1_{/.1 q}c1_{/.1 q}c2_/ c3 X t D1  ₁ 1 acqn_´ 1 1 ac´ 

C qc2_{.1 q}c3 c2_{/.1 q}c2_{/.1 q}c3_/ c2 c1 X t D1  ₁ 1 acqnCc1_´ 1 1 acqc1_´  : Consequently after converting it back into the original form,

n X t D1 1 P8.W; W / D ˛ 2mC4 A2_B4_U 2km1U2km2U2km3U2k.m3 m2/U2k.m2 m1/U2k.m3 m1/   U2k.m1Cm2 m3/U2km3U2k.m2 m1/  2km2 X t D1  ˛nC2tk W_nC2tkC2m.A2_{; B}2_/ ˛2t k W_{2t kC2m}.A2_{; B}2_/  U2k.m2 m1/U2km1U2km2 2km3 X t D1  ˛nC2tk W_nC2tkC2m.A2_{; B}2_/ ˛2t k W_{2t kC2m}.A2_{; B}2_/  C U2k.m3 m2/U2km2U2km3  2k.m2 m1/ X t D1  ˛nC2tkC2km1 W_{nC2tkC2km1C2m}.A2_{; B}2_/ ˛2t kC2km1 W_{2t kC2km1C2m}.A2_{; B}2_/  : 3. ii) We now consider the sums

n X t D1 U_{2k.t Cg/C2m}U_{2k.t C2g/C2m} P8.W; W ; g; 2g; 3g/ :

Converting it into q-form and rewriting the parameters, that is, ´D q2t k, q2mD c, q2kg _{D q}d and .B=A/2D a, we get

A 8˛6 6gk 4m.1 q/2

n

X

t D1

´.1 c´qd/.1 c´q2d/

.1 ac´/.1 ac´qd_{/.1 ac´q}2d_{/.1 ac´q}3d_/:

Without constant factor, we consider SnWD

n

X

t D1

´.1 c´qd/.1 c´q2d/

.1 ac´/.1 ac´qd_{/.1 ac´q}2d_{/.1 ac´q}3d_/:

By partial fraction decomposition of the summand of Snand using telescoping, we

obtain

SnD

1

  .a qd/.1 a/.1 q3d/ d X t D1  1 1 ac´qn 1 1 ac´  .1 aqd/.1 aq2d/.1 qd/ 3d X t D1  1 1 ac´qn 1 1 ac´  C .1 aqd/.1 a/.1 q3d/ 2d X t D1  1 1 ac´qn 1 1 ac´  : Thus we get n X t D1 U_{2k.t Cg/C2m}U_{2k.t C2g/C2m} P8.W; W ; g; 2g; 3g/ D 1 B6_A4_U2 2kgU4kg  W2kg.B2; A2/.A2 B2/U6kg  d X t D1  ˛2mCn 6gkC2ktC4 W_nC2ktC2m.A2_{; B}2_/ ˛2m 6gkC2ktC4 W_{2k t C2m}.A2_{; B}2_/ 
W2kg.A2; B2/W4kg.A2; B2/U2kg  3d X t D1  ˛2mCn 6gkC2ktC4 W_nC2ktC2m.A2_{; B}2_/ ˛2m 6gkC2ktC4 W_{2k t C2m}.A2_{; B}2_/  C W2kg.A2; B2/.A2 B2/U6kg  2d X t D1  ˛2mCn 6gkC2ktC4 W_nC2ktC2m.A2_{; B}2_/ ˛2m 6gkC2ktC4 W_{2k t C2m}.A2_{; B}2_/  ; where
is defined as before.

4. i) For 0 < m1< m2< m3< m4, we consider the sums n X t D1 U_{2k.t Cm1/C2m}U_{2k.t Cm2/C2m}U_{2k.t Cm3/C2m} P10.W; W / :

If we convert it into q-notation and write ´D q2t k, q2m D c, q2kmi _{D q}ci _and .B=A/2_{D a for 1  i  4, then the sums above equals}

A 10˛7 2km4 4m_{.1 q/}2  n X t D1 ´.1 c´qc1_{/.1 c´q}c2_{/.1 c´q}c3_/

Without constant factor, we consider the sums SnWD n X t D1 ´.1 c´qc1_{/.1 c´q}c2_{/.1 c´q}c3_/

.1 ac´/.1 ac´qc1_{/.1 ac´q}c2_{/.1 ac´q}c3_{/.1 ac´q}c4_/: The partial fraction decomposition of the summand is

´.1 c´qc1_{/.1 c´q}c2_{/.1 c´q}c3_/a4_c

.1 ac´/.1 ac´qc1_{/.1 ac´q}c2_{/.1 ac´q}c3_{/.1 ac´q}c4_/

D B 1 1 acqc1_´ 1 1 ac´  C 1 1 acqc2_´ 1 1 ac´  C D_{1 acq}1 c3´ 1 1 ac´  E 1 1 acqc4´ 1 1 ac´  ; where A_Da 3_{.1 a} 1_qc1_{/.1 a} 1_qc2_{/.1 a} 1_qc3_/ .1 qc1_{/.1 q}c2_{/.1 q}c3_{/.1 q}c4_/ ; B_D a 2_{.1 a} 1_qc2 c1_{/.1 a} 1_qc3 c1_{/.1 a/} .1 qc1/.1 qc2 c1/.1 qc3 c1/.1 qc4 c1/; C _D a.1 aq c2 c1_{/.1 a} 1_qc3 c2_{/.1 a/} .1 qc2/.1 qc2 c1/.1 qc3 c2/.1 qc4 c2/; DD .1 aq c3 c1_{/.1 aq}c3 c2_{/.1 a/} .1 qc3_{/.1 q}c3 c1_{/.1 q}c3 c2_{/.1 q}c4 c3_/; E_D .1 aq c4 c1_{/.1 aq}c4 c2_{/.1 aq}c4 c3_/ .1 qc4_{/.1 q}c4 c1_{/.1 q}c4 c2_{/.1 q}c4 c3_/I note that AD C C E B D.

Then by telescoping and converting the sums into the original notion and taking care again about the omitted factor, we obtain

n X t D1 U_{2k.t Cm1/C2m}U_{2k.t Cm2/C2m}U_{2k.t Cm3/C2m} P10.W; W / D.A 2 _B2_/˛2mC5
A6_B8 _W 2k.m2 m1/.B 2_{; A}2_/W 2k.m3 m1/.B 2_{; A}2_/ U2km1U2k.m2 m1/U2k.m3 m1/U2k.m4 m1/ T1 CW2k.m2 m1/.A 2_{; B}2_/W 2k.m3 m2/.B 2_{; A}2_/ U2km2U2k.m2 m1/U2k.m3 m2/U2k.m4 m2/ T2 CW2k.m3 m1/.A 2_{; B}2_/W 2k.m3 m2/.A 2_{; B}2_/ U2km3U2k.m3 m1/U2k.m3 m2/U2k.m4 m3/ T3
W2k.m4 m2/.A 2_{; B}2_/W 2k.m4 m1/.A 2_{; B}2_/W 2k.m4 m3/.A 2_{; B}2_/ .A2 _B2_/U 2km4U2k.m4 m1/U2k.m4 m2/U2k.m4 m3/ T4  ;

with ˛ ˇD
and the sums Ti D 2kmi X t D1  ˛nC2tk W_nC2tkC2m.A2; B2/ ˛2t k W_{2t kC2m}.A2; B2/  : 4. ii) For 0 < m1< m2< m3< m4< m5, we consider the sums

n X t D1 1 P12.W; W / :

If we convert it into q-notation and write ´D q2t k, q2m D c, q2kmi _{D q}ci _and .B=A/2_{D a for 1  i  5; then it equals}

.1 q/6 A12_˛2.6mCkm1Ckm2Ckm3Ckm4Ckm5 6/  n X t D1 ´3

.1 ac´/ .1 ac´qc1_{/ .1 ac´q}c2_{/ .1 ac´q}c3_{/ .1 ac´q}c4_{/ .1 ac´q}c5_/: Without constant factor, we consider the sums

SnWD n

X

t D1

 _´3

.1 ac´/ .1 ac´qc1_{/ .1 ac´q}c2_/

 1

.1 ac´qc3_{/ .1 ac´q}c4_{/ .1 ac´q}c5_/ 

:

By partial fraction decomposition of the summand and using telescoping, we write a3c3SnD F n X t D1  1 1 acqc5_´ 1 1 ac´  C E n X t D1  1 1 acqc4_´ 1 1 ac´  D n X t D1  1 1 acqc3_´ 1 1 ac´  C C n X t D1  1 1 acqc2_´ 1 1 ac´  B n X t D1  1 1 acqc1_´ 1 1 ac´  ; where AD 1 .1 qc1_{/.1 q}c2_{/.1 q}c3_{/.1 q}c4_{/.1 q}c5_/; B_D q 2c1 .1 qc1_/.qc1 _qc2_/.qc1 _qc3_/.qc1 _qc4_/.qc1 _qc5_/;

C _D q 2c2 .1 qc2_/.qc1 _qc2_/.qc2 _qc3_/.qc2 _qc4_/.qc2 _qc5_/; D_D q 2c3 .1 qc3_/.qc1 _qc3_/.qc2 _qc3_/.qc3 _qc4_/.qc3 _qc5_/; E_D q 2c4 .1 qc4_/.qc1 _qc4_/.qc2 _qc4_/.qc3 _qc4_/.qc4 _qc5_/; F _D q 2c5 .1 qc5_/.qc1 _qc5_/.qc2 _qc5_/.qc3 _qc5_/.qc4 _qc5_/I note that AD F EC D C C B. Consequently, we obtain n X t D1 1 P12.W; W /D ˛6
A6_B6   ₁ U2km5U2k.m5 m1/U2k.m5 m2/U2k.m5 m3/U2k.m5 m4/ T5 C_U 1 2km3U2k.m4 m1/U2k.m4 m2/U2k.m4 m3/U2k.m5 m4/ T4 1 U2km3U2k.m3 m1/U2k.m3 m2/U2k.m4 m3/U2k.m5 m3/ T3 C 1 U2km2U2k.m2 m1/U2k.m3 m2/U2k.m4 m2/U2k.m5 m2/ T2 1 U2km1U2k.m2 m1/U2k.m3 m1/U2k.m4 m1/U2k.m5 m1/ T1  ; where the sum Ti is defined as before.

Of course, we could invent many more examples, but we think that the message is clear now.

It should be noted that our elementary method can always be used to simplify sums of the type considered here; even if they do not telescope, they lead to simpler answers.

REFERENCES

[1] R. Melham, “Finite reciprocal sums involving summands that are balanced products of generalized Fibonacci numbers.” J. Integer Seq., vol. 17, no. 6, pp. article 14.6.5, 11, 2014.

Authors’ addresses

E. Kılıc¸

TOBB University of Economics and Technology, Mathematics Department, 06560 Ankara, Turkey E-mail address: ekilic@etu.edu.tr

H. Prodinger

Department of Mathematics, University of Stellenbosch, 7602 Stellenbosch, South Africa E-mail address: hproding@sun.ac.za