05A30 Keywords: reciprocal sums identities, partial fraction decomposition 1

Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 20 (2019), No. 2, pp. 1039–1050 DOI: 10.18514/MMN.2019.2810

NEW RECIPROCAL SUMS INVOLVING FINITE PRODUCTS OF SECOND ORDER RECURSIONS

EMRAH KILIC¸ AND DIDEM ERSANLI Received 15 January, 2019

Abstract. In this paper, we present new kinds of reciprocal sums of finite products of general second order linear recurrences. In order to evaluate explicitly them byq-calculus, first we convert them into theirq-notation and then use the methods of partial fraction decomposition and creative telescoping.

2010Mathematics Subject Classification: 11B37; 05A30

Keywords: reciprocal sums identities, partial fraction decomposition

1. INTRODUCTION

Forn > 1, define the second order linear recurrencesfUngandfVngwith UnDpUn 1CUn 2andVnDpVn 1CVn 2;

whereU0D0,U1D1andV0D2,V1Dp, respectively.

IfpD1, thenUnDFn(nth Fibonacci number) andVnDLn(nth Lucas number).

IfpD2, thenUnDPn(nth Pell number) andVnDQn(nth Pell-Lucas number).

The Binet forms are UnD˛ⁿ ˇⁿ

˛ ˇ D˛^{n 1}.1 qⁿ/

.1 q/ andVnD˛ⁿCˇⁿD˛ⁿ.1Cqⁿ/;

where˛; ˇD.p˙p

p²C4/=2,qDˇ=˛andiDp 1.

Throughout this paper we will use the following notations: the q-Pochhammer symbol.xIq/nD.1 x/.1 xq/ : : : .1 xq^{n 1}/. WhenxDq;we denote.qIq/nby .q/n:

Many authors [1–11] have studied both finite or infinite and alternating or non- alternating reciprocal sums including terms of certain integer sequences. More re- cently, Frontczak [3] evaluated various reciprocal sums of the Fibonacci numbers.

For example, he showed that form; n1

N

X

iD1

. 1/^m.iC1/ FmiCn1

F_{m.i 1/Cn}FmiCnF_m.iC1/Cn D Fm1

FnFmF2m

c 2019 Miskolc University Press

F_m.NC1/

F_m.NC1/CnC FmN

FmNCn

Fm

FmCn

. 1/^m FmN

F_m²FmCnF_m.NC1/Cn: Kılıc¸ and Prodinger [5] consider some classes of reciprocal sums of general Fibon- acci numbers, which were computed in closed form in an earlier work and then they evaluate the same sums by a different approach: First they convert them into their q-forms and then explicitly evaluateq-versions of the sums by using partial fraction decomposition method and creative telescoping idea.

In this paper, we shall consider new kinds of reciprocal sums including finite products of the general Fibonacci and Lucas numbers. We shall summarize what we will present in this paper below.

We consider three kinds ofalternatingreciprocal sums including finite products of the general Fibonacci and Lucas numbers. All of them have an integer parameter to increase or decrease the indices of the general Fibonacci or Lu- cas factors in both numerator and denominator in the sums. Two of these sums are of the forms

n

X

kD0

. 1/^k U_{k d}

U_kCdU_kCdC1U_kCdC2 and

n

X

kD0

. 1/^k V_kCdC1

U_kCdU_kCdC1U_kCdC2:

The other kind sums has an additional parameter m which determines the number of the general Fibonacci or Lucas factors in the numerator or de- nominator. It will be of the form

n

X

kD0

. 1/^k U_kCcU_kCcC1: : : U_{kCcCm 1} X_kCdX_kCdC1: : : X_kCdCmC1;

whereXnis eitherUnorVn:

Before all, we convert all the claimed results into theirq-forms. After this, we will use partial fraction decomposition (pfd) method and creative telescoping idea to prove theq-version of the claimed results.

By theq-versions of three kinds of reciprocal sums, we present general cases of the original three kinds of sums with an additional integer parameter by taking a special choosing of q. By the way, we could able to derive non- alternating reciprocal sums where the indices of the general Fibonacci or Lucas numbers are in the arithmetic progressions.

Finally we shall give some applications of our results to certain alternating and non-alternating reciprocal sums with finite products of the Pell or Pell- Lucas numbers.

2. THE MAIN RESULTS

Now we are going to present our main results. We start with our first main result.

Theorem 1. Forn; m0andc2 f mC1; : : : ; 1; 0; 1g;

n

X

kD0

. 1/^k U_kCcU_kCcC1: : : U_{kCcCm 1} X_kCdX_kCdC1: : : X_kCdCmC1

D. 1/^cC1 UnCcUnCcC1: : : UnCcCm

UmC1Xd cC1ŒXnCdC1XnCdC2: : : XnCdCmC1; where X_n is eitherU_n orV_n: Note that when X_nDV_n, we assume that d is any integer. WhenXnDUn, we assume thatd 1.

Proof. LetXnDUn. First we convert the LHS of the claim into itsq-notation as shown

n

X

kD0

. 1/^k

m 1

Q

tD0

UkCtCc mC1

Q

tD0

U_kCtCd

D˛^{mc .mC2/d 2mC1}.1 q/²

n

X

kD0

q^k

m 1

Q

tD0

1 q^kCcCt

mC1

Q

tD0

.1 q^kCdCt/ :

Second we convert the RHS of the claim into itsq-notation as shown

. 1/^cC1

m

Q

tD0

UnCtCc

UmC1U_{d cC1}

mC1

Q

tD1

U_nCtCd

D. 1/^cC1˛^{.mC2/c .mC2/d 2m 1}

.1 q/²

m

Q

tD0

1 q^nCcCt

.1 q^mC1/.1 q^{d cC1}/

mC1

Q

tD1

.1 q^nCdCt/ :

After some simplifications, theq-version of the claimed result is

n

X

kD0

q^k

m 1

Q

tD0

1 q^kCcCt

mC1

Q

tD0

.1 q^kCdCt/ D

q^{1 c}

m

Q

tD0

1 q^nCcCt

.1 q^mC1/.1 q^{d cC1}/

mC1

Q

tD1

.1 q^nCdCt/

or, in terms of theq-Pochhammer notation,

n

X

kD0

q^k

q^kCcIq

m

q^kCdIq

mC2

D q^{1 c} q^nCcIq

mC1

.1 q^mC1/.1 q^{d cC1}/ q^nCdC1Iq

mC1

:

Define

SnWD

n

X

kD0

q^k

q^kCcIq

m

q^kCdIq

mC2

:

Denote the summand term ofSnbyT_k, that is, T_kD

q^k

q^kCcIq

m

q^kCdIq

mC2

:

The partial fraction decomposition ofT_k reads as q^k

q^kCcIq

m

q^kCdIq

mC2

D

mC2

X

tD1

q ^{d tC1}

q^{c d tC1}Iq

m

.1 q^{kCdCt 1}/

mC2

Q

iD1 i6Dt

.1 q^{i t}/ :

From [5], we could obtain the following identity comes from creative telescoping idea:

n

X

kD0

1 1 q^kCdCm

1 1 q^kCdCt

D

m t 1

X

kD0

1

1 q^kCdCnCtC1

1 1 q^kCdCt

:

(2.1) In that case, by using the equality (2.1), we write

SnD

mC1

X

tD1

1 1 q^{dCt 1}

1 1 q^dCnCt

^t X

rD1

q ^{d rC1}

q^{c d rC1}Iq

m

q^{1 r}Iq

r 1.q/_{m rC2}

D. 1/^m 1 1 q^mC1

mC1

X

tD1

1 1 q^{dCt 1}

1 1 q^dCnCt

q^{.m tC1/c .m tC2/dCt .t 1/Cm.m22tC1/}

q^{c d 2}Iq

3 t

q^{d c}Iq

t mC1

.q/_{t 1}.q/_{m tC1}

D. 1/^m1 q^nC1 1 q^mC1

mC1

X

tD1

q^{t . cCd mCt /} .1 q^{dCt 1}/.1 q^dCnCt/

q^{c 1 dC12m.mC1C2c 2d /}

q^{c d 2}Iq

3 t

q^{d c}Iq

t mC1

.q/_{t 1}.q/_{m tC1}

D 1 q^nC1

.1 q^{c d 1}/.1 q^mC1/

mC1

X

tD1

. 1/^tC1

q.^tC21/ ¹

q^{c d 1}Iq

2 t

q^{c d}Iq

m tC1

.1 q^{dCt 1}/.1 q^dCnCt/ .q/_{t 1}.q/_{m tC1}

D 1 q^nC1

.1 q^{c d 1}/.1 q^mC1/

m

X

tD0

. 1/^t

q.^tC22 / ¹

q^{c d 1}Iq

1 t

q^{c d}Iq

m t

.1 q^dCt/.1 q^dCnCtC1/ .q/_t.q/_{m t}

D 1 q^nC1

.1 q^mC1/.1 q^{c d 1}/

m

X

tD0

. 1/^t q.^t2/

.q ^t q^d/.q ^t q^dCnC1/

.1 q^{c d t 1}/.1 q^{c d t}/ : : : .1 q^{c dCm t 2}/.1 q^{c dCm t 1}/

.q/_t.q/_{m t} :

Now we define

h.´/WD .1 ´q^{c d 1}/ : : : .1 ´q^{c dCm 1}/

.´ q^d/.´ q^dCnC1/.1 ´/.1 ´q/ : : : .1 ´q^m/: The partial fraction decomposition ofh.´/is

h.´/ D .1 q^{c 1}/ : : : .1 q^{cCm 1}/

.´ q^d/.q^d q^dCnC1/.1 q^d/.1 q^dC1/ : : : .1 q^dCm/

C .1 q^cCn/ : : : .1 q^cCnCm/

.q^dCnC1 q^d/.´ q^dCnC1/.1 q^dCnC1/.1 q^dCnC2/ : : : .1 q^dCnCmC1/ C

m

X

tD0

. 1/^tq.^tC21/ .1 q ^{tCc d 1}/ : : : .1 q ^{tCc dCm 1}/ .q ^t q^d/.q ^t q^dCnC1/ .q/_t.q/_{m t}.1 ´q^t/:

If we multiplyh.´/by´and let´! 1, then we get 0D q ^d.1 q^{c 1}/ : : : .1 q^{cCm 1}/

.1 q^nC1/.1 q^d/.1 q^dC1/ : : : .1 q^dCm/ q ^d.1 q^cCn/ : : : .1 q^cCnCm/

.1 q^nC1/.1 q^dCnC1/.1 q^dCnC2/ : : : .1 q^dCnCmC1/

C

m

X

tD0

. 1/^tC1q.^2t/ .1 q ^{tCc d 1}/ : : : .1 q ^{tCc dCm 1}/ .q ^t q^d/.q ^t q^dCnC1/ .q/_t.q/_{m t} :

Sincec2 f mC1; : : : ; 1; 0; 1g;we write

m

X

tD0

. 1/^tq.^2t/ .1 q ^{tCc d 1}/ : : : .1 q ^{tCc dCm 1}/ .q ^t q^d/.q ^t q^dCnC1/ .q/_t.q/_{m t}

D q ^d.1 q^nCc/ : : : .1 q^cCmCn/

.1 q^nC1/.1 q^dCnC1/.1 q^dCnC2/ : : : .1 q^dCmCnC1/: Taking into the constant factor, we write

1 q^nC1

.1 q^mC1/.1 q^{c d 1}/

m

X

tD0

. 1/^tq.^2t/C1.1 ´q^{c d 1}/ : : : .1 ´q^{c dCm 1}/ .´ q^d/.´ q^dCnC1/ .q/_t.q/_{m t}

D q ^d.1 q^nCc/ : : : .1 q^cCmCn/

.1 q^mC1/.1 q^{c d 1}/.1 q^dCnC1/ : : : .1 q^dCmCnC1/

D q ^d q^nCcIq

mC1

.1 q^mC1/.1 q^{c d 1}/ q^nCdC1Iq

mC1

D q^{1 c} q^nCcIq

mC1

.1 q^mC1/.1 q^{d cC1}/ q^nCdC1Iq

mC1

;

which completes the proof.

Also, whenXnDVn;the proof is similarly obtained.

Now we will present some interesting corollaries of Theorem 1. When mD2;

XnDLn,UnDFn; dD3andcD0in Theorem1, it gives us

n

X

kD0

. 1/^k FkFkC1

LkC3LkC4LkC5LkC6 D FnFnC1FnC2

F3L4LnC4LnC5LnC6

:

WhenmD3; XnDUnDPn; dD5andcD1in Theorem1, we get

n

X

kD0

. 1/^k P_kC1P_kC2P_kC3

P_kC5P_kC6P_kC7P_kC8P_kC9 D PnC1PnC2PnC3PnC4

P4P5PnC6PnC7PnC8PnC9

:

WhenmD4; XnDQn,UnDPn; d D4andcD 2in Theorem1, we get

n

X

kD0

. 1/^k P_{k 2}P_{k 1}P_kP_kC1

Q_kC4Q_kC5Q_kC6Q_kC7Q_kC8Q_kC9 D

P_{n 2}P_{n 1}P_nP_nC1P_nC2 P5Q7QnC5QnC6QnC7QnC8QnC9

:

WhenmD5; XnDUnDFn; d D5andcD 3in Theorem1, we get

n

X

kD0

. 1/^k F_{k 3}F_{k 2}F_{k 1}F_kF_kC1

F_kC5F_kC6F_kC7F_kC8F_kC9F_kC10F_kC11

D Fn 3Fn 2Fn 1FnFnC1FnC2

F6F9FnC6FnC7FnC8FnC9FnC10FnC11

:

Now we shall give our second result. The first main result stands for the finite product of the general Fibonacci or Lucas numbers. But the next two results are valid for special cases.

Theorem 2. Ford > 0,

n

X

kD0

. 1/^k U_{k d}

U_kCdU_kCdC1U_kCdC2 D. 1/^dC1 U2nC2

U2U_nCdC1U_nCdC2: (2.2) Proof. First we convert the LHS of (2.2) into itsq-notation as shown

n

X

kD0

. 1/^k U_{k d}

U_kCdU_kCdC1U_kCdC2

D˛ ^{4d 1}.1 q/²

n

X

kD0

q^k.1 q^{k d}/

.1 q^kCd/.1 q^kCdC1/.1 q^kCdC2/: Second we convert the RHS of (2.2) into itsq-notation as shown

. 1/^dC1 U2nC2

U₂U_nCdC1U_nCdC2D˛ ^{2d 1}.1 q/² . 1/^dC1.1 q^2nC2/ 1 q²

.1 q^dCnC1/.1 q^dCnC2/: After some simplifications,q-version of the claimed result is

n

X

kD0

q^k.1 q^{k d}/

.1 q^kCd/.1 q^kCdC1/.1 q^kCdC2/D q ^d.1 q^2nC2/ 1 q²

.1 q^dCnC1/.1 q^dCnC2/: Define

SnWD

n

X

kD0

´.1 ´q ^d/

.1 ´q^d/.1 ´q^dC1/.1 ´q^dC2/: Denote the summand term ofSnbyT .´/, that is,

T .´/D ´.1 ´q ^d/

.1 ´q^d/.1 ´q^dC1/.1 ´q^dC2/: The partial fraction decomposition ofT .´/reads as

´.1 ´q ^d/

.1 ´q^d/.1 ´q^dC1/.1 ´q^dC2/

D 1

q^3dC1.1 q/².1Cq/

q.1 q^2d/

1 ´q^d C.1Cq/.1 q^2dC1/ 1 ´q^dC1

1 q^2dC2 1 ´q^dC2

!

D 1

q^3dC1.1 q/².1Cq/

q.1 q^2d/

1 1 ´q^dC2

1 1 ´q^d

.1Cq/.1 q^2dC1/

1 1 ´q^dC2

1 1 ´q^dC1

:

And so

SnD 1

q^3dC1.1 q/².1Cq/

"

q.1 q^2d/

n

X

kD0

1 1 ´q^dC2

1 1 ´q^d

(2.3)

.1Cq/.1 q^2dC1/

n

X

kD0

1 1 ´q^dC2

1 1 ´q^dC1

# :

If we takemD2,tD0;and,mD2,tD1in (2.1), respectively, then we rewriteSn

given in (2.3) as 1

q^3dC1.1 q/².1Cq/

"

q.1 q^2d/

1

X

kD0

1 1 q^kCdC1Cn

1 1 q^kCd

.1Cq/.1 q^2dC1/

1 1 q^dCnC2

1 1 q^dC1

D

1Cq^nC1

1 q^dC1

.1 q^nC1/ q^d 1 q^dC1

1 q^dCnC1

1 q^dCnC2

.1 q/ .1Cq/

D q ^d.1 q^2nC2/ 1 q²

.1 q^dCnC1/.1 q^dCnC2/:

Thus, we have the conclusion.

WhenUnDFnanddD3in (2.2), we obtain

n

X

kD0

. 1/^k Fk 3

F_kC3F_kC4F_kC5 D F2nC2

FnC4FnC5

:

Now we going to give our third result:

Theorem 3. Ford > 0,

n

X

kD0

. 1/^k V_kCdC1

U_kCdU_kCdC1U_kCdC2 D UnC1UnC2.dC1/

U1U_dU_dC1U_nCdC1U_nCdC2: (2.4) Proof. After required converting and simplifications, we find theq-version of the claimed result as follows

n

X

kD0

q^k.1Cq^kCdC1/

.1 q^kCd/.1 q^kCdC1/.1 q^kCdC2/

D .1 q^nC1/.1 q^nC2.dC1// .1 q/ 1 q^d

1 q^dC1

.1 q^nCdC1/.1 q^nCdC2/:

Define

SnWD

n

X

kD0

´.1C´q^dC1/

.1 ´q^d/.1 ´q^dC1/.1 ´q^dC2/: Denote the summand term ofSnbyT .´/, that is,

T .´/D ´.1C´q^dC1/

.1 ´q^d/.1 ´q^dC1/.1 ´q^dC2/: The partial fraction decomposition ofT .´/reads as

´.1C´q^dC1/

.1 ´q^d/.1 ´q^dC1/.1 ´q^dC2/

D 1

q^d.1Cq/.1 q/²

1Cq 1 ´q^d

2.1Cq/

1 ´q^dC1C 1Cq 1 ´q^dC2

D 1

q^d.1Cq/.1 q/²

.1Cq/

1 1 ´q^dC2

1 1 ´q^d

2.1Cq/

1 1 ´q^dC2

1 1 ´q^dC1

:

And so

SnD 1

q^d.1Cq/.1 q/²

"

.1Cq/

n

X

kD0

1 1 ´q^dC2

1 1 ´q^d

2.1Cq/

n

X

kD0

1 1 ´q^dC2

1 1 ´q^dC1

# :

If we takemD2,tD0;and,mD2,tD1in (2.1), respectively, then we rewrite the last equality as

SnD 1

q^d.1Cq/.1 q/²

"

.1Cq/

1

X

kD0

1 1 q^kCdC1Cn

1 1 q^kCd

2.1Cq/

1 1 q^dCnC2

1 1 q^dC1

D .1 q^nC1/.1 q^nC2dC2/

.1 q/.1 q^d/.1 q^dC1/.1 q^nCdC1/.1 q^nCdC2/:

Thus, we have the conclusion.

WhenUnDFn; VnDLnandd D5in (2.4), we obtain

n

X

kD0

. 1/^k L_kC6

F_kC5F_kC6F_kC7 D FnC1FnC12

F5F6FnC6FnC7

:

3. GENERAL CASES

In the previous section, we found theq-versions of the claimed results in Theorems 1-3 while proving them. Now we shall present more general cases of Theorems 1-3 by taking q Dˇ^s=˛^s for any integer s in the q-forms of them without proof, respectively.

Theorem 4. Forn; m0andc2 f mC1; : : : ; 1; 0; 1g;

n

X

kD0

. 1/^sk

m 1

Q

tD0

U_{s.kCcCt /}

mC1

Q

tD0

Xs.kCdCt /

D. 1/^s.cC1/

m

Q

tD0

U_s.nCtCc/

Us.mC1/Xs.d cC1/

mC1

Q

tD1

Xs.nCtCd /

;

(3.1) where Xn is eitherUn orVn: Note that when XnDVn, we assume that d is any integer. WhenXnDUn, we assume thatd 1.

In (3.1), if we takemD3; sD2; UnDPn; XnDQn; d D4andcD1;then we obtain

n

X

kD0

P_2kC2P_2kC4P_2kC6

Q_2kC8Q_2kC10Q_2kC12Q_2kC14Q_2kC16D P2nC2P2nC4P2nC6P2nC8

P8Q8Q2nC10Q2nC12Q2nC14Q2nC16

:

WhenmD4; sD 1; UnDXnDFn; d D6andcD0in Theorem4, then, by F nD. 1/^nC1Fn;we obtain

n

X

kD0

. 1/^k F _kF _{k 1}F _{k 2}F _{k 3}

F _{k 6}F _{k 7}F _{k 8}F _{k 9}F _{k 10}F _{k 11}

D

n

X

kD0

. 1/^k F_kF_kC1F_kC2F_kC3

F_kC6F_kC7F_kC8F_kC9F_kC10F_kC11

D FnFnC1FnC2FnC3FnC4

F5F7FnC7FnC8FnC9FnC10FnC11

:

WhenmD5; sD3; UnDFn; XnDLn; d D2andcD 3in Theorem 4, then we obtain

n

X

kD0

. 1/^k F_{3k 9}F_{3k 6}F_{3k 3}F_3kF_3kC3

L_3kC6L_3kC9L_3kC12L_3kC15L_3kC18L_3kC21L_3kC24

D F3n 9F3n 6F3n 3F3nF3nC3F3nC6

F18L18L3nC9L3nC12L3nC15L3nC18L3nC21L3nC24

:

Theorem 5. Ford > 0,

n

X

kD0

. 1/^sk U_{s.k d /}

Us.kCd /Us.kCdC1/Us.kCdC2/ D. 1/^sdC1 U_s.2nC2/

U2sUs.nCdC1/Us.nCdC2/

:

(3.2) If we takeUnDFn; dD2andsD5in (3.2), then we obtain

n

X

kD0

. 1/^k F_{5k 10}

F_5kC10F_5kC15F_5kC20 D F10nC10

F10F5nC15F5nC20

:

Theorem 6. Ford > 0,

n

X

kD0

. 1/^sk V_s.kCdC1/

U_{s.kCd /}U_s.kCdC1/U_s.kCdC2/ D U_s.nC1/U_s.nC2dC2/

UsU_sdU_s.dC1/U_s.nCdC1/U_s.nCdC2/: (3.3) If we take UnDPn; Vn DQn; d D4 and s D 3 in (3.3), then, by P nD . 1/^nC1PnandQ nD. 1/ⁿQn;we obtain

n

X

kD0

. 1/^k Q _{3k 15}

P _{3k 12}P _{3k 15}P _{3k 18} D

n

X

kD0

. 1/^k Q_3kC15 P_3kC12P_3kC15P_3kC18

D P3nC3P3nC30

P3P12P15P3nC15P3nC18

:

4. CONCLUSIONS

In the last section, we derived more general results, where the sign functions and the indices of Fibonacci or Lucas factors are depend on the integer parameters. By the way, one can see that these generalized reciprocal sums are alternating for odd integers;and, non-alternating for even integerswhile the original sums in Theorems 1-3are always alternating sums.

REFERENCES

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Authors’ addresses

Emrah Kılıc¸

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

Didem Ersanlı

TOBB University of Economics and Technology, Mathematics Department, 06560, Ankara, Turkey E-mail address:didemersanli@gmail.com