Some Computational Formulas for 𝐷-Nörlund Numbers

Abstract

The author establishes some identities involving the 𝐷 numbers, Bernoulli numbers, and central factorial numbers of the first kind. A generating function and several computational formulas for 𝐷-Nörlund numbers are also presented.

1. Introduction and Results

The Bernoulli polynomials 𝐵_𝑛^(𝑘)(𝑥) of order 𝑘, for any integer 𝑘, may be defined by (see [1–5]) (𝑡𝑒^𝑡)−1^𝑘𝑒^𝑥𝑡=^∞_𝑛=0𝐵_𝑛^(𝑘)𝑡(𝑥)^𝑛𝑛!,|𝑡|<2𝜋. (1.1)

The numbers 𝐵_𝑛^(𝑘)=𝐵_𝑛^(𝑘)(0) are the Bernoulli numbers of order 𝑘, 𝐵_𝑛⁽¹⁾=𝐵_𝑛 are the ordinary Bernoulli numbers (see [2, 6, 7]). By (1.1), we can get (see [4, page 145]) 𝑑𝐵𝑑𝑥^𝑛(𝑘)(𝑥)=𝑛𝐵^(𝑘)_𝑛−1(𝑥),𝐵_𝑛^(𝑘+1)(𝑥)=𝑘−𝑛𝑘𝐵_𝑛^(𝑘)𝑛(𝑥)+(𝑥−𝑘)𝑘𝐵^(𝑘)_𝑛−1(𝑥),𝐵_𝑛^(𝑘+1)(𝑥+1)=𝑛𝑥𝑘𝐵^(𝑘)_𝑛−1(𝑥)−𝑛−𝑘𝑘𝐵_𝑛^(𝑘)(𝑥), (1.2)(1.3)(1.4) where 𝑛∈ℕ, with ℕ being the set of positive integers.

The numbers 𝐵_𝑛^(𝑛) are called the Nörlund numbers (see [4, 8]). A generating function for the Nörlund numbers 𝐵_𝑛^(𝑛) is (see [4, page 150]) 𝑡=(1+𝑡)log(1+𝑡)^∞_𝑛=0𝐵_𝑛^(𝑛)𝑡^𝑛.𝑛! (1.5)

The 𝐷 numbers 𝐷^(𝑘)_2𝑛 may be defined by (see [4, 5]) (𝑡csc𝑡)^𝑘=^∞_𝑛=0(−1)^𝑛𝐷^(𝑘)_2𝑛𝑡^2𝑛(2𝑛)!,|𝑡|<𝜋. (1.6)

By (1.1), (1.6), and note that csc𝑡=2𝑖/(𝑒^𝑖𝑡−𝑒^−𝑖𝑡) (where 𝑖²=−1), we can get 𝐷^(𝑘)_2𝑛=4^𝑛𝐵^(𝑘)_2𝑛(𝑘2). (1.7)

Taking 𝑘=1,2 in (1.7), and note that 𝐵⁽¹⁾_2𝑛(1/2)=(2^1−2𝑛−1)𝐵_2𝑛, 𝐵⁽²⁾_2𝑛(1)=(1−2𝑛)𝐵_2𝑛 (see [4, pages 22 and 145]), we have 𝐷⁽¹⁾_2𝑛=(2−2^2𝑛)𝐵_2𝑛,𝐷⁽²⁾_2𝑛=4^𝑛(1−2𝑛)𝐵_2𝑛. (1.8)

The numbers 𝐷^(2𝑛)_2𝑛 are called the 𝐷-Nörlund numbers. These numbers 𝐷^(2𝑛)_2𝑛 and 𝐷^(2𝑛−1)_2𝑛 have many important applications. For example (see [4, page 246]) ₀^𝜋/2sin𝑡𝑡𝑑𝑡=^∞_𝑛=0(−1)^𝑛𝐷^(2𝑛)_2𝑛,(2𝑛+1)!₀^𝜋/2sin𝑡𝑡𝜋𝑑𝑡=2^∞_𝑛=0(−1)^𝑛+1𝐷^(2𝑛−1)_2𝑛2^2𝑛(2𝑛−1)(𝑛!)²,2𝜋=^∞_𝑛=0(−1)^𝑛+1𝐷^(2𝑛−1)_2𝑛.(2𝑛−1)(2𝑛)! (1.9)(1.10)

We now turn to the central factorial numbers 𝑡(𝑛,𝑘) of the first kind, which are usually defined by (see [9–12]) 𝑛𝑥(𝑥+2𝑛−1)(𝑥+2𝑛−2)⋯(𝑥+2−𝑛+1)=^𝑛_𝑘=0𝑡(𝑛,𝑘)𝑥^𝑘, (1.11) or by means of the following generating function: 𝑥(2log(2+𝑥1+²4))^𝑘=𝑘!^∞_𝑛=𝑘𝑥𝑡(𝑛,𝑘)^𝑛.𝑛! (1.12)

It follows from (1.11) or (1.12) that 1𝑡(𝑛,𝑘)=𝑡(𝑛−2,𝑘−2)−4(𝑛−2)²𝑡(𝑛−2,𝑘), (1.13) and that 𝑡(𝑛,0)=𝛿_𝑛,0(𝑛∈ℕ₀∶=ℕ∪{0}),𝑡(𝑛,𝑛)=1(𝑛∈ℕ),𝑡(𝑛,𝑘)=0(𝑛+𝑘odd),𝑡(𝑛,𝑘)=0(𝑘>𝑛or𝑘<0), (1.14) where 𝛿_𝑚,𝑛 denotes the Kronecker symbol.

By (1.13), we have 𝑡(2𝑛+1,1)=(−1)^𝑛(2𝑛)!4^2𝑛𝑛2𝑛,𝑡(2𝑛+2,2)=(−1)^𝑛(𝑛!)²(𝑛∈ℕ₀),𝑡(2𝑛+2,4)=(−1)^𝑛+1(𝑛!)²1(1+2²+13²1+⋯+𝑛²)(𝑛∈ℕ),𝑡(2𝑛+1,3)=(−1)^𝑛−1(2𝑛)!4^2𝑛−1𝑛2𝑛1(1+3²+15²1+⋯+(2𝑛−1)²)(𝑛∈ℕ). (1.15)(1.16)(1.17)

The main purpose of this paper is to prove some identities involving 𝐷 numbers, Bernoulli numbers, and central factorial numbers of the first kind and obtain a generating function and several computational formulas for the 𝐷-Nörlund numbers. That is, we will prove the following main conclusion.

Theorem 1.1. Let 𝑛∈ℕ, 𝑘∈ℕ⧵{1}. Then 𝐷^(𝑘)_2𝑛=(2𝑛−𝑘+2)(2𝑛−𝑘+1)𝐷(𝑘−2)(𝑘−1)^{(𝑘−2)2𝑛}−2𝑛(2𝑛−1)(𝑘−2)𝐷𝑘−1^{(𝑘−2)2𝑛−2}. (1.18)

Remark 1.2. By (1.18), we may immediately deduce the following (see [4, page 147]: 𝐷^(2𝑛+1)_2𝑛=(−1)^𝑛(2𝑛)!4^𝑛𝑛2𝑛,𝐷^(2𝑛+2)_2𝑛=(−1)^𝑛4^𝑛2𝑛+1(𝑛!)². (1.19)

Theorem 1.3. Let 𝑛≥𝑘(𝑛,𝑘∈ℕ₀). Then 𝐷^(2𝑛+1)_{2𝑛−2𝑘}=4^𝑛−𝑘(^2𝑛_2𝑘)𝑡(2𝑛+1,2𝑘+1),𝐷^(2𝑛)_{2𝑛−2𝑘}=4^𝑛−𝑘(^2𝑛−1_2𝑘−1)𝑡(2𝑛,2𝑘)(𝑘≥1). (1.20)(1.21)

Remark 1.4. By (1.20) and (1.17), we may immediately deduce the following: 𝐷^(2𝑛+3)_2𝑛=(−1)^𝑛(2𝑛)!2⋅4^2𝑛2𝑛+21𝑛+1(1+3²+15²1+⋯+(2𝑛+1)²). (1.22)

Theorem 1.5. Let 𝑛∈ℕ₀. Then ^𝑛_𝑗=0(−1)^𝑗4^𝑗𝑗𝐷(2𝑗+1)2𝑗^{(2𝑛−2𝑗)}_{2𝑛−2𝑗}=(2𝑛−2𝑗)!(−1)^𝑛4^𝑛𝑛,2𝑛 (1.23) so one finds 𝐷₀⁽⁰⁾=1,𝐷₂⁽²⁾=−2/3,𝐷₄⁽⁴⁾=88/15,𝐷₆⁽⁶⁾=−3056/21,𝐷₈⁽⁸⁾=319616/45,𝐷⁽¹⁰⁾₁₀=−18940160/33,….
By (1.23), and note that
log(𝑡+√1+𝑡²)=^∞_𝑛=0(−1)^𝑛4^𝑛𝑛𝑡(2𝑛+1)2𝑛_2𝑛+11(|𝑡|<1),√1+𝑡²=^∞_𝑛=0(−1)^𝑛4^𝑛𝑛𝑡2𝑛_2𝑛(|𝑡|<1), (1.24) one may immediately deduce the following Corollary 1.6.

Corollary 1.6. Let 𝑛∈ℕ₀. Then ^∞_𝑛=0𝐷^(2𝑛)_2𝑛𝑡^2𝑛=𝑡(2𝑛)!√1+𝑡²log(𝑡+√1+𝑡²)(|𝑡|<1). (1.25)

Theorem 1.7. Let 𝑛∈ℕ. Then(i)𝐷^(2𝑛)_2𝑛=(−1)^𝑛(2𝑛)!4^𝑛𝑛2𝑛+𝑛⋅4^𝑛𝑛_𝑗=1(1−2^1−2𝑗)𝑗𝑡(2𝑛,2𝑗)𝐵_2𝑗 (1.26)(ii)𝐷^(2𝑛)_2𝑛=(−1)^𝑛4^𝑛(𝑛!)²2𝑛+1+𝑛⋅4^𝑛𝑛_𝑗=12𝑗−1𝑗𝑡(2𝑛,2𝑗)𝐵_2𝑗 (1.27)(iii)𝐷^(2𝑛)_2𝑛=^𝑛_𝑗=04^𝑛−𝑗2𝑗+1𝑡(2𝑛+1,2𝑗+1). (1.28)

Theorem 1.8. Let 𝑛∈ℕ₀. Then(i)^𝑛_𝑗=02−4^𝑗4^𝑗𝑡(2𝑛+1,2𝑗+1)𝐵_2𝑗=(−1)^𝑛(𝑛!)²2𝑛+1 (1.29)(ii)^𝑛_𝑗=0(1−2𝑗)𝑡(2𝑛+1,2𝑗+1)𝐵_2𝑗=(−1)^𝑛(2𝑛)!2⋅4^2𝑛2𝑛+21𝑛+1(1+3²+15²1+⋯+(2𝑛+1)²). (1.30)

2. Proof of the Theorems

Proof of Theorem 1.1. By (1.4) and (1.3), we have 𝐵^(𝑘)_2𝑛(𝑥+1)=2𝑛𝑥𝐵𝑘−1^{(𝑘−1)2𝑛−1}(𝑥)−2𝑛−𝑘+1𝐵𝑘−1^{(𝑘−1)2𝑛}=(𝑥)2𝑛𝑥(𝑘−1𝑘−2𝑛−1𝐵𝑘−2^{(𝑘−2)2𝑛−1}(𝑥)+(𝑥−𝑘+2)2𝑛−1𝐵𝑘−2^{(𝑘−2)2𝑛−2}−(𝑥))2𝑛−𝑘+1(𝑘−1𝑘−2𝑛−2𝐵𝑘−2^{(𝑘−2)2𝑛}(𝑥)+(𝑥−𝑘+2)2𝑛𝐵𝑘−2^{(𝑘−2)2𝑛−1}=(𝑥))(2𝑛−𝑘+1)(2𝑛−𝑘+2)𝐵(𝑘−1)(𝑘−2)^{(𝑘−2)2𝑛}(𝑥)+2𝑛(2𝑛−1)(𝑘−1)(𝑘−2)𝑥(𝑥−𝑘+2)𝐵^(𝑘−2)_2𝑛−2−(𝑥)2𝑛(2𝑛−𝑘+1)(𝑘−1)(𝑘−2)(2𝑥−𝑘+2)𝐵^(𝑘−2)_2𝑛−1(𝑥). (2.1) Setting 𝑥=(𝑘−2)/2 in (2.1), we get 𝐵^(𝑘)_2𝑛(𝑘2)=(2𝑛−𝑘+1)(2𝑛−𝑘+2)𝐵(𝑘−1)(𝑘−2)^{(𝑘−2)2𝑛}(𝑘−22)−2𝑛(2𝑛−1)(𝑘−2)𝐵4(𝑘−1)^{(𝑘−2)2𝑛−2}(𝑘−22). (2.2) By (2.2) and (1.7), we immediately obtain (1.18). This completes the proof of Theorem 1.1.

Proof of Theorem 1.3. By the usage of Theorem 1.1 and (1.13).

Proof of Theorem 1.5. Note the identity (see [4, page 203]) 𝐵^(𝑘)_2𝑛+1𝑘(𝑥+2)=^𝑛_𝑗=02𝑛+1𝐷2𝑗+12^{(𝑘−2𝑗−1)2𝑛−2𝑗2𝑛−2𝑗}𝑥(𝑥²1−(2)²)(𝑥²3−(2)²)⋯(𝑥²−(2𝑗−12)²), (2.3) we have lim_𝑥→0𝐵^(2𝑛+1)_2𝑛+1(𝑥+(2𝑛+1)/2)𝑥=14^𝑛𝑛_𝑗=0𝐷2𝑛+12𝑗+1^{(2𝑛−2𝑗)2𝑛−2𝑗}(−1)^𝑗1²⋅3²⋯(2𝑗−1)²=(2𝑛+1)!4^𝑛𝑛_𝑗=0(−1)^𝑗4^𝑗𝑗𝐷(2𝑗+1)2𝑗^{(2𝑛−2𝑗)}_{2𝑛−2𝑗}.(2𝑛−2𝑗)! (2.4) By (2.4) and (1.2), we have lim_𝑥→0(2𝑛+1)𝐵^(2𝑛+1)_2𝑛(𝑥+2𝑛+12)=(2𝑛+1)!4^𝑛𝑛_𝑗=0(−1)^𝑗4^𝑗𝑗𝐷(2𝑗+1)2𝑗^{(2𝑛−2𝑗)}_{2𝑛−2𝑗},(2𝑛−2𝑗)! (2.5) that is, 𝐵^(2𝑛+1)_2𝑛(2𝑛+12)=(2𝑛)!4^𝑛𝑛_𝑗=0(−1)^𝑗4^𝑗𝑗𝐷(2𝑗+1)2𝑗^{(2𝑛−2𝑗)}_{2𝑛−2𝑗}.(2𝑛−2𝑗)! (2.6) By (2.6) and (1.7), we have 𝐷^(2𝑛+1)_2𝑛=(2𝑛)!^𝑛_𝑗=0(−1)^𝑗4^𝑗𝑗𝐷(2𝑗+1)2𝑗^{(2𝑛−2𝑗)}_{2𝑛−2𝑗}.(2𝑛−2𝑗)! (2.7) By (2.7) and (1.19), we immediately obtain (1.23). This completes the proof of Theorem 1.5.

Proof of Theorem 1.7. By (1.6), we have 𝐷^(𝑘)_2𝑛=^𝑛_𝑗=0𝐷2𝑛2𝑗^{(𝑘−𝑙)2𝑛−2𝑗}𝐷^(𝑙)_2𝑗, (2.8) where 𝑙 is an integer.
Setting 𝑘=2𝑛+1,𝑙=1 in (2.8), and note that 𝐷₀⁽¹⁾=1, we have
𝐷^(2𝑛+1)_2𝑛=^𝑛_𝑗=0𝐷2𝑛2𝑗^{(2𝑛)2𝑛−2𝑗}𝐷⁽¹⁾_2𝑗=𝐷^(2𝑛)_2𝑛+^𝑛_𝑗=1𝐷2𝑛2𝑗^{(2𝑛)2𝑛−2𝑗}𝐷⁽¹⁾_2𝑗. (2.9) By (2.9), (1.19), (1.8), and (1.21), we immediately obtain (1.26).
Setting 𝑘=2𝑛+2,𝑙=2 in (2.8), and note that 𝐷₀⁽²⁾=1, we have
𝐷^(2𝑛+2)_2𝑛=^𝑛_𝑗=0𝐷2𝑛2𝑗^{(2𝑛)2𝑛−2𝑗}𝐷⁽²⁾_2𝑗=𝐷^(2𝑛)_2𝑛+^𝑛_𝑗=1𝐷2𝑛2𝑗^{(2𝑛)2𝑛−2𝑗}𝐷⁽²⁾_2𝑗. (2.10) By (2.10), (1.19), (1.8), and (1.21), we immediately obtain (1.27).
Setting 𝑘=2𝑛,𝑙=−1 in (2.8), and note that (1.20) and 𝐷⁽⁻¹⁾_2𝑗=1/(2𝑗+1), we immediately obtain (1.18). This completes the proof of Theorem 1.7.

Proof of Theorem 1.8. Setting 𝑘=2𝑛+2,𝑙=1 in (2.8), and note (1.19), (1.20), and (1.8), we immediately obtain (1.29).
Setting 𝑘=2𝑛+3,𝑙=2 in (2.8), and note (1.22), (1.20), and (1.8), we immediately obtain (1.30). This completes the proof of Theorem 1.8.

Acknowledgment

This work was supported by the Guangdong Provincial Natural Science Foundation (no. 8151601501000002).