On Reciprocal Series of Generalized Fibonacci Numbers with Subscripts in Arithmetic Progression

Abstract

We investigate formulas for closely related series of the forms: ∑^∞_𝑛=01/(𝑈_{𝑎𝑛+𝑏}+𝑐), ∑^∞_𝑛=0(−1)^𝑛𝑈_{𝑎𝑛+𝑏}/(𝑈_{𝑎𝑛+𝑏}+𝑐)², ∑^∞_𝑛=0𝑈_{2(𝑎𝑛+𝑏)}/(𝑈²_{𝑎𝑛+𝑏}+𝑐)² for certain values of 𝑎, 𝑏, and 𝑐.

1. Introduction

Let 𝑝 be a nonzero integer such that Δ=𝑝²+4≠0. The generalized Fibonacci and Lucas sequences are defined by the following recurrences:𝑈_𝑛+1=𝑝𝑈_𝑛+𝑈_𝑛−1,𝑉_𝑛+1=𝑝𝑉_𝑛+𝑉_𝑛−1, (1.1) where 𝑈₀=0,𝑈₁=1 and 𝑉₀=2,𝑉₁=𝑝, respectively. When 𝑝=1,𝑈_𝑛=𝐹_𝑛 (𝑛th Fibonacci number) and 𝑉_𝑛=𝐿_𝑛 (𝑛th Lucas number).

If 𝛼 and 𝛽 are the roots of equation 𝑥²−𝑝𝑥−1=0, the Binet formulas of the sequences {𝑈_𝑛} and {𝑉_𝑛} have the forms:𝑈_𝑛=𝛼^𝑛−𝛽^𝑛𝛼−𝛽,𝑉_𝑛=𝛼^𝑛+𝛽^𝑛, (1.2) respectively.

In [1], Backstrom developed formulas for closely related series of the form:^∞_𝑛=01𝐹_{𝑎𝑛+𝑏}+𝑐, (1.3) for certain values of 𝑎, 𝑏, and 𝑐. For example, he obtained the following series:^∞_𝑛=01𝐹_2𝑛+1+𝐹_𝐾=𝐾√52𝐿_{𝐾∞𝑛=0},1𝐹_{(2𝑛+1)𝐾+2𝑡}+𝐹_𝐾=√5−5𝐹_𝑡/𝐿_𝑡2𝐿_𝐾√,𝑡even,5−𝐿_𝑡/𝐹_𝑡2𝐿_𝐾,𝑡odd, (1.4) where 𝐾 represents an odd integer and 𝑡 is an integer in the range −(𝐾−1)/2 to (𝐾−1)/2 inclusive. Also, he gave the similar results for Lucas numbers.

In [2], Popov found in explicit form series of the form:^∞_𝑛=01𝐹_{𝑎𝑛+𝑏},±𝑐^∞_𝑛=01𝐹_{𝑎𝑛+𝑏}𝐹_{𝑐𝑛+𝑑},^∞_𝑛=01𝐹²_{𝑎𝑛+𝑏}±𝐹²_{𝑐𝑛+𝑑}, (1.5) for certain values of 𝑎, 𝑏, 𝑐, and 𝑑.

In [3], Popov generalized some formulas of Backstrom [1] related to sums of reciprocal series of Fibonacci and Lucas numbers. For example, Δ^∞_𝑛=0−𝑞^{𝑠+𝑛𝑟}𝑉_{(2𝑛+1)𝑟+2𝑠}−−𝑞^{𝑠+𝑛𝑟}𝑉_𝑟=𝛽^𝑠𝑈_𝑟𝑈_𝑠,𝛽𝛼𝛼<1,_𝑠𝑈_𝑟𝑈_𝑠,𝛼𝛽<1, (1.6) where 𝑠 and 𝑟 are integers.

In [4], Gauthier found the closed form expressions for the following sums:^𝑚_𝑘=1(−1)^𝑘𝑛𝑓_{(2𝑘+1)𝑛}𝑓²_()𝑛𝑘+1𝑓²_𝑘𝑛,𝑚,𝑛≥1,_𝑚𝑘=0(−1)^𝑘𝑛𝑓_{(2𝑘+1)𝑛}𝑙²_()𝑛𝑘+1𝑙²_𝑘𝑛,𝑚,𝑛≥0, (1.7) where for 𝑥≠0 an indeterminate, the generalized Fibonacci and Lucas polynomials {𝑓_𝑛}_𝑛 and {𝑙_𝑛}_𝑛 are given by the following recurrences:𝑓_𝑛+2=𝑥𝑓_𝑛+1+𝑓_𝑛,𝑓₀=0,𝑓₁𝑙=1,𝑛≥0,_𝑛+2=𝑥𝑙_𝑛+1+𝑙_𝑛,𝑙₀=2,𝑙₁=𝑥,𝑛≥0, (1.8) respectively.

In this paper, we investigate formulas for closely related series of the forms:^∞_𝑛=01𝑈_{𝑎𝑛+𝑏},+𝑐^∞_𝑛=0(−1)^𝑛𝑈_{𝑎𝑛+𝑏}𝑈_{𝑎𝑛+𝑏}+𝑐²,^∞_𝑛=0𝑈_{2(𝑎𝑛+𝑏)}𝑈²_{𝑎𝑛+𝑏}+𝑐², (1.9) for certain values of 𝑎, 𝑏 and 𝑐.

2. On Some Series of Reciprocals of Generalized Fibonacci Numbers

In this section, firstly, we will give the following lemmas for further use.

Lemma 2.1. Let 𝑛 be an arbitrary nonzero integer. For integer 𝑚≥1, ^𝑚_𝑘=1(−1)^𝑘𝑛𝑈_{(2𝑘+1)𝑛}𝑈²_()𝑛𝑘+1𝑈²_𝑘𝑛=14𝑈_𝑛𝑉²_𝑛𝑈²_𝑛−𝑉²_()𝑛𝑚+1𝑈²_()𝑛𝑚+1, (2.1) and for integer 𝑚≥0, ^𝑚_𝑘=0(−1)^𝑘𝑛𝑈_{(2𝑘+1)𝑛}𝑉²_()𝑛𝑘+1𝑉²_𝑘𝑛=𝑈²_()𝑛𝑚+14𝑈_𝑛𝑉²_()𝑛𝑚+1. (2.2)

Proof. We give the proof of Lemma 2.1 as the proofs of the sums in [4], using the following equalities: 𝑈_{(2𝑘+1)𝑛}𝑈_(𝑘+1)𝑛𝑈_𝑘𝑛=12𝑉_𝑘𝑛𝑈_𝑘𝑛+𝑉_(𝑘+1)𝑛𝑈_(𝑘+1)𝑛,𝑈_{(2𝑘+1)𝑛}𝑉_(𝑘+1)𝑛𝑉_𝑘𝑛=12𝑈_(𝑘+1)𝑛𝑉_(𝑘+1)𝑛+𝑈_𝑘𝑛𝑉_𝑘𝑛,(−1)^𝑘𝑛𝑈_𝑛𝑈_(𝑘+1)𝑛𝑈_𝑘𝑛=12𝑉_𝑘𝑛𝑈_𝑘𝑛−𝑉_(𝑘+1)𝑛𝑈_(𝑘+1)𝑛,(−1)^𝑘𝑛𝑈_𝑛𝑉_(𝑘+1)𝑛𝑉_𝑘𝑛=12𝑈_(𝑘+1)𝑛𝑉_(𝑘+1)𝑛−𝑈_𝑘𝑛𝑉_𝑘𝑛. (2.3)

Lemma 2.2. For arbitrary integers 𝑛 and 𝑡, 𝑉_2𝑛−(−1)^𝑛−𝑡𝑉_2𝑡=Δ𝑈_𝑛−𝑡𝑈_𝑛+𝑡,𝑉_2𝑛+(−1)^𝑛−𝑡𝑉_2𝑡=𝑉_𝑛−𝑡𝑉_𝑛+𝑡,𝑈²_𝑛−(−1)^𝑛−𝑡𝑈²_𝑡=𝑈_𝑛−𝑡𝑈_𝑛+𝑡,𝑉²_𝑛−(−1)^𝑛−𝑡𝑉²_𝑡=Δ𝑈_𝑛−𝑡𝑈_𝑛+𝑡. (2.4)(2.5)

Proof. From Binet formulas of sequences {𝑈_𝑛} and {𝑉_𝑛}, the desired results are obtained.

Theorem 2.3. For an odd integer 𝑡, ^𝑚_𝑛=11𝑈_{(2𝑛+1)𝑡}+𝑈_𝑡=12𝑉_𝑡2−𝑉_2𝑡𝑈_2𝑡−2−𝑉_{2(𝑚+1)𝑡}𝑈_{2(𝑚+1)𝑡𝑚𝑛=1},1𝑈_{(2𝑛+1)𝑡}−𝑈_𝑡=12𝑉_𝑡2+𝑉_2𝑡𝑈_2𝑡−2+𝑉_{2(𝑚+1)𝑡}𝑈_{2(𝑚+1)𝑡}. (2.6)

Proof. By replacing 𝑛 with (2𝑛+1)𝑡 in (2.5), we have 𝑈²_{()𝑡2𝑛+1}−𝑈²_𝑡=𝑈_2𝑛𝑡𝑈_{2(𝑛+1)𝑡}, (2.7) or 1𝑈_{(2𝑛+1)𝑡}+𝑈_𝑡=𝑈_{(2𝑛+1)𝑡}−𝑈_𝑡𝑈_2𝑛𝑡𝑈_{2(𝑛+1)𝑡}. (2.8) Taking 𝑟=(2𝑛+1)𝑡 and 𝑠=𝑡 in the equality 𝑉_𝑠𝑈_𝑟=𝑈_𝑟+𝑠+(−1)^𝑠𝑈_𝑟−𝑠 [5], the equality (2.8) is rewritten as follows: 1𝑈_{(2𝑛+1)𝑡}+𝑈_𝑡=1𝑉_𝑡1𝑈_2𝑛𝑡+(−1)^𝑡𝑈_{2(𝑛+1)𝑡}−𝑈_𝑡𝑈_2𝑛𝑡𝑈_{2(𝑛+1)𝑡}. (2.9) We have the sum ^𝑚_𝑛=11𝑈_{(2𝑛+1)𝑡}+𝑈_𝑡=1𝑉_𝑡^𝑚_𝑛=11𝑈_2𝑛𝑡+(−1)^𝑡𝑈_{2(𝑛+1)𝑡}−𝑈_𝑡^𝑚_𝑛=11𝑈_2𝑛𝑡𝑈_{2(𝑛+1)𝑡}. (2.10) For an odd integer 𝑡, we have ^𝑚_𝑛=11𝑈_2𝑛𝑡−1𝑈_{2(𝑛+1)𝑡}=1𝑈_2𝑡−1𝑈_{2(𝑚+1)𝑡}, (2.11) and taking 𝑠=2𝑛𝑡 and 𝑟=2𝑡 in identity [5]: 𝑈_𝑠+𝑟𝑉_𝑠−𝑈_𝑠𝑉_𝑠+𝑟=2(−1)^𝑠𝑈_𝑟, (2.12) we get ^𝑚_𝑛=11𝑈_2𝑛𝑡𝑈_{2(𝑛+1)𝑡}=12𝑈^𝑚_2𝑡𝑛=1𝑉_2𝑛𝑡𝑈_2𝑛𝑡−𝑉_{2(𝑛+1)𝑡}𝑈_{2(𝑛+1)𝑡}=12𝑈_2𝑡𝑉_2𝑡𝑈_2𝑡−𝑉_{2(𝑚+1)𝑡}𝑈_{2(𝑚+1)𝑡}. (2.13)
Substituting (2.11) and (2.13) in (2.10), we have the desired result.

For example, if we take 𝑡=1 and 𝑝=1 in (2.6), we have ^𝑚_𝑛=11𝐹_2𝑛+1=𝐹+1^2𝑚+1−1𝐹_2(𝑚+1). (2.14) Note that 𝐹^𝑚_{2(𝑚+1)𝑛=1}1𝐹_2𝑛+1=+1^𝑚_𝑛=1𝐹_2𝑛. (2.15)

Corollary 2.4. For an odd integer 𝑡, ^𝑚_𝑛=11𝑈_{(2𝑛+1)𝑡}+𝑈_𝑡=12𝑉_𝑡𝑉_(𝑚+1)𝑡𝑈_(𝑚+1)𝑡−𝑉_𝑡𝑈_𝑡1,𝑚iseven,2𝑉_𝑡Δ𝑈_(𝑚+1)𝑡𝑉_(𝑚+1)𝑡−𝑉_𝑡𝑈_𝑡,𝑚isodd,_𝑚𝑛∑₌₁1𝑈_{(2𝑛+1)𝑡}−𝑈_𝑡=Δ2𝑉_𝑡𝑈_𝑡𝑉_𝑡−𝑈_(𝑚+1)𝑡𝑉_(𝑚+1)𝑡1,𝑚iseven,2𝑉_𝑡Δ𝑈_𝑡𝑉_𝑡−𝑉_(𝑚+1)𝑡𝑈_(𝑚+1)𝑡,𝑚isodd. (2.16)

Proof. Using the equalities 𝑉_2𝑛=𝑉²_𝑛−2(−1)^𝑛=Δ𝑈²_𝑛+2(−1)^𝑛 and 𝑈_2𝑛=𝑈_𝑛𝑉_𝑛 in Theorem 2.3, the results are obtained.

Corollary 2.5. Let 𝑡 be an odd integer. For |𝛽/𝛼|<1,𝑡>0 and |𝛼/𝛽|<1,𝑡<0, ^∞_𝑛=11𝑈_{(2𝑛+1)𝑡}+𝑈_𝑡=12𝑉_𝑡√Δ−𝑉_𝑡𝑈_{𝑡∞𝑛=1},1𝑈_{(2𝑛+1)𝑡}−𝑈_𝑡=Δ2𝑉_𝑡𝑈_𝑡𝑉_𝑡−1√Δ, (2.17) and for |𝛽/𝛼|<1,𝑡<0 and |𝛼/𝛽|<1,𝑡>0, ^∞_𝑛=11𝑈_{(2𝑛+1)𝑡}+𝑈_𝑡=−12𝑉_𝑡√Δ+𝑉_𝑡𝑈_{𝑡∞𝑛=1},1𝑈_{(2𝑛+1)𝑡}−𝑈_𝑡=Δ2𝑉_𝑡𝑈_𝑡𝑉_𝑡+1√Δ. (2.18)

Proof. Since lim_𝑛→∞𝑉_{𝑎𝑛+𝑏}𝑈_{𝑎𝑛+𝑏}=√Δ𝛽𝛼𝛼<1,𝑎>0,𝛽−√<1,𝑎<0,Δ𝛽𝛼𝛼<1,𝑎<0,𝛽<1,𝑎>0, (2.19) the results are easily seen by equalities (2.16).

Theorem 2.6. For an integer 𝑚≥1 and an arbitrary nonzero integer 𝑡, ^𝑚_𝑛=1(−1)^𝑛𝑡𝑈_{(2𝑛+1)𝑡}𝑉_{(2𝑛+1)𝑡}−(−1)^𝑛𝑡𝑉_𝑡²=14Δ²𝑈_𝑡𝑉²_𝑡𝑈²_𝑡−𝑉²_()𝑡𝑚+1𝑈²_()𝑡𝑚+1. (2.20)

Proof. By replacing 𝑛 with (2𝑛+1)𝑡/2 and 𝑡 with 𝑡/2 in (2.4), we have 𝑉_{(2𝑛+1)𝑡}−(−1)^𝑛𝑡𝑉_𝑡=Δ𝑈_𝑛𝑡𝑈_(𝑛+1)𝑡, (2.21) or 1𝑉_{(2𝑛+1)𝑡}−(−1)^𝑛𝑡𝑉_𝑡=1Δ𝑈_𝑛𝑡𝑈_(𝑛+1)𝑡. (2.22) Multiplying equality (2.22) by (−1)^𝑛𝑡𝑈_{(2𝑛+1)𝑡}/𝑈_𝑛𝑡𝑈_(𝑛+1)𝑡, we get (−1)^𝑛𝑡𝑈_{(2𝑛+1)𝑡}𝑈_𝑛𝑡𝑈_(𝑛+1)𝑡𝑉_{(2𝑛+1)𝑡}−(−1)^𝑛𝑡𝑉_𝑡=(−1)^𝑛𝑡𝑈_{(2𝑛+1)𝑡}Δ𝑈²_𝑛𝑡𝑈²_()𝑡𝑛+1. (2.23) We have the sum: ^𝑚_𝑛=1(−1)^𝑛𝑡𝑈_{(2𝑛+1)𝑡}𝑈_𝑛𝑡𝑈_(𝑛+1)𝑡𝑉_{(2𝑛+1)𝑡}−(−1)^𝑛𝑡𝑉_𝑡=1Δ^𝑚_𝑛=1(−1)^𝑛𝑡𝑈_{(2𝑛+1)𝑡}𝑈²_𝑡𝑛𝑈²_𝑡()𝑛+1. (2.24) Using the equalities (2.1) and (2.21), the proof is obtained.

Corollary 2.7. For an arbitrary nonzero integer 𝑡, ^∞_𝑛=1(−1)^𝑛𝑡𝑈_{(2𝑛+1)𝑡}𝑉_{(2𝑛+1)𝑡}−(−1)^𝑛𝑡𝑉_𝑡²=14Δ²𝑈_𝑡𝑉²_𝑡𝑈²_𝑡−Δ. (2.25)

Proof. Taking 𝑚→∞ in Theorem 2.6 and using (2.19), the result is easily obtained.

Theorem 2.8. For an integer 𝑚≥0 and an arbitrary nonzero integer 𝑡, ^𝑚_𝑛=0(−1)^𝑛𝑡𝑈_{(2𝑛+1)𝑡}𝑉_{(2𝑛+1)𝑡}+(−1)^𝑛𝑡𝑉_𝑡²=𝑈²_()𝑡𝑚+14𝑈_𝑡𝑉²_()𝑡𝑚+1. (2.26)

Proof. The proof of the theorem is similar to the proof of Theorem 2.6.

Corollary 2.9. For an arbitrary nonzero integer 𝑡, ^∞_𝑛=0(−1)^𝑛𝑡𝑈_{(2𝑛+1)𝑡}𝑉_{(2𝑛+1)𝑡}+(−1)^𝑛𝑡𝑉_𝑡²=14Δ𝑈_𝑡. (2.27)

Proof. Taking 𝑚→∞ in Theorem 2.8 and using (2.19), the result is easily obtained.

For example, if we take 𝑡=3 and 𝑝=1 in (2.27), we have ^∞_𝑛=0(−1)^𝑛𝐹_3(2𝑛+1)𝐿_3(2𝑛+1)+(−1)^𝑛4²=1.40 (2.28)

Theorem 2.10. For an integer 𝑚≥1 and an arbitrary nonzero integer 𝑡, ^𝑚_𝑛=1𝑈_{2(2𝑛+1)𝑡}𝑈²_{()𝑡2𝑛+1}−𝑈²_𝑡²=14𝑈_2𝑡𝑉²_2𝑡𝑈²_2𝑡−𝑉²_{2()𝑡𝑚+1}𝑈²_{2()𝑡𝑚+1𝑚𝑛=1},𝑈_{2(2𝑛+1)𝑡}𝑉²_{()𝑡2𝑛+1}−𝑉²_𝑡²=14Δ²𝑈_2𝑡𝑉²_2𝑡𝑈²_2𝑡−𝑉²_{2()𝑡𝑚+1}𝑈²_{2()𝑡𝑚+1}. (2.29)

Proof. The proof of theorem is similar to the proof of Theorem 2.6.

Corollary 2.11. For an arbitrary nonzero integer 𝑡, ^∞_𝑛=1𝑈_{2(2𝑛+1)𝑡}𝑈²_{()𝑡2𝑛+1}−𝑈²_𝑡²=14𝑈_2𝑡𝑉²_2𝑡𝑈²_2𝑡−Δ_∞𝑛=1,𝑈_{2(2𝑛+1)𝑡}𝑉²_{()𝑡2𝑛+1}−𝑉²_𝑡²=14Δ²𝑈_2𝑡𝑉²_2𝑡𝑈²_2𝑡−Δ. (2.30)

Proof. Taking 𝑚→∞ in Theorem 2.10 and using (2.19), the result is easily obtained.

For example, if we take 𝑡=2 in the equality (2.30), we have ^∞_𝑛=1𝑈_4(2𝑛+1)𝑉²_2()2𝑛+1−𝑉²₂²=1Δ²𝑈³₄. (2.31)