A Note on Certain Modular Equations about Infinite Products of Ramanujan

Abstract

Ramanujan proposed additive formulae of theta functions that are related to modular equations about infinite products. Employing these formulaes, we derived some identities on infinite products. In the same spirit, we also could present elementary and simple proofs of certain Ramanujan's modular equations on infinite products.

1. Introduction

𝑄-theory is undoubtedly one of the most famous and useful mathematical theorems, such as Andrews-Askey type integral [1] ^𝑑_𝑐𝑞𝑡/𝑐,𝑞𝑡/𝑑,𝑓𝑡,𝑟𝑠𝑡;𝑞_∞𝑎𝑡,𝑏𝑡,𝑒𝑡,𝑠𝑡;𝑞_∞³𝜙₂𝑐𝑟,𝑏𝑡,𝑔𝑑𝑟𝑠𝑡,𝑏𝑐;𝑞,𝑠𝑔_𝑞𝑡=𝑑1−𝑞𝑞,𝑐/𝑑,𝑞𝑑/𝑐,𝑎𝑏𝑐𝑑,𝑑𝑓,𝑟𝑠𝑑;𝑞_∞𝑎𝑐,𝑏𝑐,𝑎𝑑,𝑏𝑑,𝑠𝑑,𝑑𝑒;𝑞_∞×^∞_𝑘=0𝑓/𝑒,𝑎𝑑,𝑠𝑑,𝑏𝑑;𝑞_𝑘(𝑐𝑒)^𝑘𝑞,𝑑𝑓,𝑟𝑠𝑑,𝑎𝑏𝑐𝑑;𝑞_𝑘×₃𝜙₂𝑟,𝑏𝑑𝑞^𝑘,𝑎𝑐𝑑𝑞^𝑘𝑔𝑟𝑠𝑑𝑞^𝑘,𝑎𝑏𝑐𝑑𝑞^𝑘;𝑞,𝑠𝑔, (1) Askey-Roy type integral [2] ^𝜋_−𝜋𝑃_𝑛𝑒^𝑖𝜃,𝑓𝑃_𝑚𝑒^𝑖𝜃×,𝑔𝜌𝑒^𝑖𝜃𝑑,𝑞𝑑𝑒^−𝑖𝜃,𝜌𝑐𝑒^−𝑖𝜃,𝑞𝑒^𝑖𝜃𝑐𝜌;𝑞_∞×𝑎𝑒^𝑖𝜃,𝑏𝑒^𝑖𝜃,𝑐𝑒^−𝑖𝜃,𝑑𝑒^−𝑖𝜃;𝑞_∞⁻¹=𝑑𝜃2𝜋𝑎𝑓;𝑞_𝑛𝑏𝑔;𝑞_𝑚𝑎𝑏𝑐𝑑,𝜌𝑐/𝑑,𝑑𝑞/𝜌𝑐,𝜌,𝑞/𝜌;𝑞_∞𝑎^𝑛𝑏^𝑚𝑞,𝑏𝑐,𝑏𝑑,𝑎𝑐,𝑎𝑑;𝑞_∞×^𝑛_𝑘=0𝑞^−𝑛,𝑎𝑐,𝑎𝑑;𝑞_𝑘𝑞^𝑘𝑞,𝑎𝑓,𝑎𝑏𝑐𝑑;𝑞_𝑘³𝜙₂𝑞^−𝑚,𝑏𝑐,𝑏𝑑𝑏𝑔,𝑎𝑏𝑐𝑑𝑞^𝑘;𝑞,𝑞, (2) Moment integrals [3] ^∞_−∞𝑃_𝑛(𝑤,𝑐)𝑃_𝑚(𝑤,𝑑)𝑎𝑤,𝑏𝑤;𝑞_∞𝑑𝛼^(𝑠,𝑡)(𝑤)=𝑎𝑐;𝑞_𝑛𝑏𝑑;𝑞_𝑚𝑎𝑏𝑠𝑡;𝑞_∞𝑎^𝑛𝑏^𝑚𝑎𝑠,𝑎𝑡,𝑏𝑠,𝑏𝑡;𝑞_∞×^𝑛_𝑘=0𝑞^−𝑛,𝑎𝑠,𝑎𝑡;𝑞_𝑘𝑞^𝑘𝑞,𝑎𝑐,𝑎𝑏𝑠𝑡;𝑞_𝑘³𝜙₂𝑞^−𝑚,𝑏𝑠,𝑏𝑡𝑏𝑑,𝑎𝑏𝑠𝑡𝑞^𝑘;𝑞,𝑞, (3) (where𝑃_𝑛(𝑎,𝑏)=(𝑎−𝑏)⋯(𝑎−𝑏𝑞^𝑛−1)),𝑞-Fractional Calculus Equations [4] and𝑞-Calculus [5]. For more information, please refer to [1–5].

The theta functions are very useful tool in researching 𝑞-series, especially in dealing with the form of the equation similar to above formulas, whose left-hand side is summation and right-hand side is integral. The additive identities of theta are one of the important of Ramanujan's contributions. Using it, we gave elementary and simple proofs of certain Ramanujan's modular equations on infinite products. For more information, please refer to [1–7].

In his notebook [8, pages 34–38], Ramanujan defines the following theta functions: 𝑓(𝑎,𝑏)∶=^∞_𝑛=−∞𝑎^{𝑛(𝑛+1)/2}𝑏^{𝑛(𝑛−1)/2}=(−𝑎;𝑎𝑏)_∞(−𝑏;𝑎𝑏)_∞(𝑎𝑏;𝑎𝑏)_∞,||𝜑𝑞=𝑎𝑏<1,∶=𝑓𝑞,𝑞^∞_𝑛=−∞𝑞^𝑛2=−𝑞;−𝑞_∞𝑞;−𝑞_∞,𝜓𝑞∶=𝑓𝑞,𝑞³=^∞_𝑛=1𝑞^{𝑛(𝑛−1)/2}=𝑞²;𝑞²_∞𝑞;𝑞²_∞, (4)(5)(6) where 𝑎;𝑞_∞∶=^∞_𝑛=01−𝑎𝑞^𝑛,𝑞<1, (7) sometimes written as 𝑎,𝑏,𝑐,…;𝑞_∞=𝑎;𝑞_∞𝑏;𝑞_∞𝑐;𝑞_∞⋯. (8) The infinite products are from the Jacobi triple product identity [8, page 35].

In the course of deduction, we used the following simple fact [9, 10]: 𝑎^𝑘;𝑞^𝑘_∞=𝑎,𝑎𝜔_𝑘,…,𝑎𝜔_𝑘^𝑘−1;𝑞_∞,𝑎,𝑞_∞=𝑎,𝑎𝑞,𝑎𝑞²,…,𝑎𝑞^𝑘−1;𝑞^𝑘_∞. (9) By definition of Ramanujan theta functions one can easily verify the following identities [8, page 45]: 𝑓(𝑎,𝑏)+𝑓(−𝑎,−𝑏)=2𝑓𝑎³𝑏,𝑎𝑏³,𝑓()()𝑏𝑎,𝑏−𝑓−𝑎,−𝑏=2𝑎𝑓𝑎,𝑎𝑏𝑎⁴𝑏⁴. (10) From (4), if𝑎𝑏=𝑐𝑑=𝑝, we have 𝑓(𝑎,𝑏)𝑓(𝑐,𝑑)=^∞_{𝑚,𝑛=−∞}𝑝^{(𝑚2+𝑛2)/2−(𝑚+𝑛)/2}𝑎^𝑚𝑐^𝑛. (11) Thus setting𝑚−𝑛=2𝑗and𝑚+𝑛=2𝑘, we find that 𝑓()𝑓()()𝑓()=𝑎,𝑏𝑐,𝑑+𝑓−𝑎,−𝑏−𝑐,−𝑑_{+∞𝑛,𝑚=−∞𝑚,𝑛,even}𝑝^{(𝑚2+𝑛2)/2−(𝑚+𝑛)/2}𝑎^𝑚𝑐^𝑛=2^∞_{𝑗,𝑘=−∞}𝑝^{𝑗2+𝑘2−𝑘}𝑎^𝑗+𝑘𝑐^𝑘−𝑗=2^∞_{𝑗,𝑘=−∞}𝑝^{𝑘(𝑘−1)}(𝑎𝑐)^𝑘𝑝^{𝑗(𝑗−1)}(𝑏𝑐)^−𝑗=2𝑓(𝑎𝑐,𝑏𝑑)𝑓(𝑎𝑑,𝑏𝑐). (12) Thus when𝑎𝑏=𝑐𝑑, we have 𝑓()𝑓()()𝑓()()𝑓().𝑎,𝑏𝑐,𝑑+𝑓−𝑎,−𝑏−𝑐,−𝑑=2𝑓𝑎𝑐,𝑏𝑑𝑎𝑑,𝑏𝑐 (13) Similarly we have that 𝑓()𝑓()()𝑓()𝑏𝑎,𝑏𝑐,𝑑−𝑓−𝑎,−𝑏−𝑐,−𝑑=2𝑎𝑓𝑐,𝑎𝑏𝑐𝑑𝑓𝑏𝑏/𝑐𝑑,𝑎𝑏𝑐𝑑.𝑏/𝑑 (14) The special case of these identities can be written as the following form by using Jacobian theta function [6, 7]: 2𝜃₁𝑥+𝑦∣2𝜏𝜃₁𝑥−𝑦∣2𝜏=𝜃₃𝑦∣𝜏𝜃₄(𝑥∣𝜏)−𝜃₃(𝑥∣𝜏)𝜃₄𝑦∣𝜏. (15) The authors of [6, 7] give simple proofs and very important use of it.

In the above two identities, putting𝑐=𝑎and𝑑=𝑏, we easily obtain 𝑓²(𝑎,𝑏)+𝑓²(−𝑎,−𝑏)=2𝑓𝑎²,𝑏²𝜑()𝑓𝑎𝑏,₂(𝑎,𝑏)−𝑓²()𝑏−𝑎,−𝑏=4𝑎𝑓𝑎,𝑎𝑏𝑎²𝑏²𝜓𝑎²𝑏². (16)(17)

2. Main Results

The sums and products of infinite are used in many domains of mathematics, such as Partition Functions [11–14], Fractal Geometry [9], Fractional Calculus [10], Fractal Time Series [4], and so on. Then the equations of it are concentrated by several mathematicians and engineers [15–18]. At the same time, it can be used in dynamic equations, differential equations [19], and partial differential equations [20].

This paper has two main purposes. The first is to derive the identities as follows: for|𝑞|<1, 𝑞;𝑞²_∞𝑞³;𝑞³²_∞+3−𝑞;𝑞²_∞−𝑞³;𝑞³²_∞=4𝜔𝑞;𝑞²_∞𝑞³;𝑞³²_∞𝑓𝑞𝜔,𝑞³𝜔²𝑓𝑞𝜔²,𝑞³𝜔𝑓²𝜔,𝑞𝜔²,𝑞;𝑞²_∞𝑞³;𝑞³²_∞−3−𝑞;𝑞²_∞−𝑞³;𝑞³²_∞=2𝑞;𝑞²_∞𝑞³;𝑞³²_∞𝑓𝜔²,𝑞𝜔𝑓²𝜔,𝑞𝜔²𝜑𝑞,𝑞;𝑞²_∞𝑞⁵;𝑞⁵²_∞−5−𝑞;𝑞²_∞−𝑞⁵;𝑞⁵²_∞=4𝜁𝑞;𝑞²_∞𝑞⁵;𝑞⁵²_∞×𝑓𝑞𝜁²,𝑞𝜁³𝑓𝜁,𝑞²𝜁⁴𝑓𝜁³,𝑞²𝜁²𝑓𝑞𝜁,𝑞𝜁⁴𝑓²𝜁,𝑞𝜁⁴𝑓²𝜁²,𝑞𝜁³, (18)(19)(20) in which𝜔=exp(2𝑖𝜋/3)and𝜁=exp(2𝑖𝜋/5). In the same way, we are able to give the simple and elementary proofs of the following identities of Ramanujan [8, 11, 12]: −𝑞;𝑞²_∞−𝑞³;𝑞³²_∞+𝑞;𝑞²_∞𝑞³;𝑞³²_∞=2−𝑞³;𝑞⁶²_∞𝑞³;𝑞⁶²_∞−𝑞²,−𝑞⁴;𝑞⁶_∞,−𝑞;𝑞²_∞−𝑞³;𝑞³²_∞−𝑞;𝑞²_∞𝑞³;𝑞³²_∞=4𝑞−𝑞⁶;𝑞⁶²_∞𝑞³;𝑞⁶²_∞−𝑞,−𝑞⁵;𝑞⁶_∞,−𝑞;𝑞²_∞−𝑞⁵;𝑞⁵²_∞−𝑞;𝑞²_∞𝑞⁵;𝑞⁵²_∞=4𝑞−𝑞,−𝑞²,−𝑞³,−𝑞⁴,−𝑞⁶,−𝑞⁷,−𝑞⁸,−𝑞⁹;𝑞¹⁰_∞𝑞⁵;𝑞¹⁰⁴_∞. (21)(22)(23)

3. Modular Equations of Infinite Productions

In this section, we first give the two sets refinement about the identities (18) and (20).

Theorem 1. For|𝑞|<1, 𝑞;𝑞_∞𝑞³;𝑞³_∞−𝑖√3−𝑞;𝑞_∞−𝑞³;𝑞³_∞=2𝜔𝑞;𝑞_∞𝑞³;𝑞³_∞𝑓𝑞𝜔,𝑞³𝜔²𝑓𝜔,𝑞𝜔²,𝑞;𝑞_∞𝑞³;𝑞³_∞+𝑖√3−𝑞;𝑞_∞−𝑞³;𝑞³_∞=2𝑞;𝑞_∞𝑞³;𝑞³_∞𝑓𝑞𝜔²,𝑞³𝜔𝑓𝜔,𝑞𝜔². (24)

Proof. Note that1+𝜔+𝜔²=0and𝜔−𝜔²=𝑖√3. By (9), we get that 𝑖√3−𝑞;𝑞_∞−𝑞³;𝑞³_∞÷𝑞;𝑞_∞𝑞³;𝑞³_∞=𝜔−𝜔²𝑞𝜔;𝑞_∞𝑞𝜔²;𝑞_∞−𝑞𝜔;𝑞_∞−𝑞𝜔²;𝑞_∞=𝜔(1−𝜔)(1+𝜔)𝑞𝜔;𝑞_∞𝑞𝜔²;𝑞_∞(1+𝜔)−𝑞𝜔;𝑞_∞−𝑞𝜔²;𝑞_∞=−𝜔;𝑞_∞𝑞𝜔²;𝑞_∞−𝜔;𝑞_∞−𝑞𝜔²;𝑞_∞𝑓=−−𝜔,−𝑞𝜔²𝑓𝜔,𝑞𝜔². (25) From (10), we have 𝑓𝜔,𝑞𝜔−𝑓−𝜔,−𝑞𝜔=2𝑓𝑞𝜔²,𝑞³𝜔𝑓,𝜔,𝑞𝜔+𝑓−𝜔,−𝑞𝜔=2𝜔𝑓𝑞𝜔,𝑞³𝜔². (26)(27) Dividing by𝑓(𝜔,𝑞𝜔), respectively, and then applying (25), we derive 𝑖√3−𝑞;𝑞_∞−𝑞³;𝑞³_∞÷𝑞;𝑞_∞𝑞³;𝑞³_∞𝑓−1=−1−−𝜔,−𝑞𝜔²𝑓𝜔,𝑞𝜔²𝑓=−2𝜔𝑞𝜔,𝑞³𝜔²𝑓𝜔,𝑞𝜔²,𝑖√3−𝑞;𝑞_∞−𝑞³;𝑞³_∞÷𝑞;𝑞_∞𝑞³;𝑞³_∞𝑓+1=1−−𝜔,−𝑞𝜔²𝑓𝜔,𝑞𝜔²𝑓=2𝑞𝜔²,𝑞³𝜔𝑓𝜔,𝑞𝜔². (28)(29) Multiplying by(𝑞;𝑞)_∞/(𝑞³;𝑞³)_∞, respectively, we complete the proofs of (24).

Proof of (19). Let𝛼=exp(𝑖𝜋/6); then it is easy to know that 𝜔=𝑖𝛼and𝛼+1/𝛼=−√3.
One has √3−𝑞;𝑞_∞−𝑞³;𝑞³_∞÷𝑞;𝑞_∞𝑞³;𝑞³_∞1=−𝛼+𝛼𝑞𝜔;𝑞_∞𝑞𝜔²;𝑞_∞−𝑞𝜔;𝑞_∞−𝑞𝜔²;𝑞_∞1=−𝛼1+𝛼²𝑞𝜔;𝑞_∞𝑞𝜔²;𝑞_∞−𝑞𝜔;𝑞_∞−𝑞𝜔²;𝑞_∞1=−𝛼(1−𝑖𝛼)(1+𝑖𝛼)𝑞𝜔;𝑞_∞𝑞𝜔²;𝑞_∞−𝑞𝜔;𝑞_∞−𝑞𝜔²;𝑞_∞1=−𝛼(1−𝜔)(1+𝜔)𝑞𝜔;𝑞_∞𝑞𝜔²;𝑞_∞−𝑞𝜔;𝑞_∞−𝑞𝜔²;𝑞_∞1=−𝛼(1+𝜔)²(𝜔;𝑞)_∞(𝑞𝜔²;𝑞)_∞−𝜔;𝑞_∞−𝑞𝜔²;𝑞_∞𝜔=−𝛼𝑓−𝜔,−𝑞𝜔²𝑓𝜔,𝑞𝜔²𝑓=−𝑖−𝜔,−𝑞𝜔²𝑓𝜔,𝑞𝜔². (30)
Then we obtain that 1+√3−𝑞;𝑞_∞−𝑞³;𝑞³_∞÷𝑞;𝑞_∞𝑞³;𝑞³_∞𝑓=1−𝑖−𝜔,−𝑞𝜔²𝑓𝜔,𝑞𝜔²,1−√3−𝑞;𝑞_∞−𝑞³;𝑞³_∞÷𝑞;𝑞_∞𝑞³;𝑞³_∞=1+𝑖−𝜔,−𝑞𝜔²𝑓𝜔,𝑞𝜔². (31) In (16), let𝑎=𝜔and𝑏=𝑞𝜔²then we have that 𝑓²𝜔,𝑞𝜔²+𝑓²−𝜔,−𝑞𝜔²=2𝑓𝜔²,𝑞²𝜔𝜑𝑞. (32) Dividing by𝑓²(𝜔,𝑞𝜔), respectively, we arrive at 𝑓1+²−𝜔,−𝑞𝜔²𝑓²𝜔,𝑞𝜔²=2𝑓𝜔²,𝑞²𝜔𝜑𝑞𝑓²𝜔,𝑞𝜔². (33) Multiplying (31), combining with (33), and then multiplyed by(𝑞;𝑞)²_∞/(𝑞³;𝑞³)²_∞, we are able to obtain (19).

Theorem 2. For|𝑞|<1, 𝑞;𝑞_∞𝑞⁵;𝑞⁵_∞−√5−𝑞;𝑞_∞−𝑞⁵;𝑞⁵_∞=2𝜁𝑞;𝑞_∞𝑞⁵;𝑞⁵_∞𝑓𝑞𝜁²,𝑞𝜁³𝑓𝜁,𝑞²𝜁⁴𝑓𝜁,𝑞𝜁⁴𝑓𝜁²,𝑞𝜁³,𝑞;𝑞_∞𝑞⁵;𝑞⁵_∞+√5(−𝑞;𝑞)_∞−𝑞⁵;𝑞⁵_∞=2𝑞;𝑞_∞𝑞⁵;𝑞⁵_∞𝑓𝜁³,𝑞²𝜁²𝑓𝑞𝜁,𝑞𝜁⁴𝑓𝜁,𝑞𝜁⁴𝑓𝜁²,𝑞𝜁³. (34)(35)

Proof. First we recall that𝜁⁵=1, 𝜁+𝜁⁴−𝜁²−𝜁³=√5 and 1+𝜁+𝜁²+𝜁³+𝜁⁴=0. Using (9), we have √5−𝑞;𝑞_∞−𝑞⁵;𝑞⁵_∞÷𝑞;𝑞_∞𝑞⁵;𝑞⁵_∞=𝜁+𝜁⁴−𝜁²−𝜁³×𝑞𝜁;𝑞_∞𝑞𝜁²;𝑞_∞𝑞𝜁³;𝑞_∞𝑞𝜁⁴;𝑞_∞−𝑞𝜁;𝑞_∞−𝑞𝜁²;𝑞_∞−𝑞𝜁³;𝑞_∞−𝑞𝜁⁴;𝑞_∞=𝜁(1−𝜁)1−𝜁²(1+𝜁)1+𝜁²×𝑞𝜁;𝑞_∞𝑞𝜁²;𝑞_∞𝑞𝜁³;𝑞_∞𝑞𝜁⁴;𝑞_∞×(1+𝜁)1+𝜁²−𝑞𝜁;𝑞_∞×−𝑞𝜁²;𝑞_∞−𝑞𝜁³;𝑞_∞−𝑞𝜁⁴;𝑞_∞⁻¹𝑓=−−𝜁,−𝑞𝜁⁴𝑓−𝜁²,−𝑞𝜁³𝑓𝜁,𝑞𝜁⁴𝑓𝜁²,𝑞𝜁³. (36) Then we know easily that 1+√5−𝑞;𝑞_∞−𝑞⁵;𝑞⁵_∞÷𝑞;𝑞_∞𝑞⁵;𝑞⁵_∞𝑓=1−−𝜁,−𝑞𝜁⁴𝑓−𝜁²,−𝑞𝜁³𝑓𝜁,𝑞𝜁⁴𝑓𝜁²,𝑞𝜁³,1−√5−𝑞;𝑞_∞−𝑞⁵;𝑞⁵_∞÷𝑞;𝑞_∞𝑞⁵;𝑞⁵_∞𝑓=1+−𝜁,−𝑞𝜁⁴𝑓−𝜁²,−𝑞𝜁³𝑓𝜁,𝑞𝜁⁴𝑓𝜁²,𝑞𝜁³. (37) In (13) and (14), setting𝑎=𝜁, 𝑏=𝑞𝜁⁴, 𝑐=𝜁², and𝑑=𝑞𝜁³, we get that 𝑓𝜁,𝑞𝜁⁴𝑓𝜁²,𝑞𝜁³+𝑓−𝜁,−𝑞𝜁⁴𝑓−𝜁²,−𝑞𝜁³=2𝑓𝜁³,𝑞²𝜁²𝑓𝑞𝜁⁴,𝑓,𝑞𝜁𝜁,𝑞𝜁⁴𝑓𝜁²,𝑞𝜁³−𝑓−𝜁,−𝑞𝜁⁴𝑓−𝜁²,−𝑞𝜁³=2𝜁𝑓𝑞𝜁²,𝑞𝜁³𝑓𝜁,𝑞²𝜁⁴. (38)
Dividing the above two equations by𝑓(𝜁,𝑞𝜁⁴)𝑓(𝜁²,𝑞𝜁³), respectively, and then combining with (37), we obtain that 1+√5−𝑞;𝑞_∞−𝑞⁵;𝑞⁵_∞÷𝑞;𝑞_∞𝑞⁵;𝑞⁵_∞𝑓𝜁=2³,𝑞²𝜁²𝑓𝑞𝜁,𝑞𝜁⁴𝑓𝜁,𝑞𝜁⁴𝑓𝜁²,𝑞𝜁³√,1−5−𝑞;𝑞_∞−𝑞⁵;𝑞⁵_∞÷𝑞;𝑞_∞𝑞⁵;𝑞⁵_∞𝑓=2𝜁𝑞𝜁²,𝑞𝜁³𝑓𝜁,𝑞²𝜁⁴𝑓𝜁,𝑞𝜁⁴𝑓𝜁²,𝑞𝜁³. (39)(40) Multiplied by(𝑞;𝑞)_∞/(𝑞⁵;𝑞⁵)_∞, the identities (39) and (40) become (34) and (35).
Multiplying the two refinements in Theorems 1 and 2, respectively, we obtain the identities (18) and (20). Using the same method, we can obtain refinement identities of (21) and (22) which are similar to Theorems 1 and 2; then we can deduce (21), (22), and (23) easily. The details of proofs are omitted.

The following conclusion can be obtained easily.

Corollary 3. For |𝑞|<1, 𝑞;𝑞⁴_∞𝑞³;𝑞³⁴_∞−9−𝑞;𝑞⁴_∞−𝑞³;𝑞³⁴_∞=8𝜔𝑞;𝑞⁴_∞𝑞³;𝑞³⁴_∞×𝑓𝜔²,𝑞𝜔𝑓𝑞𝜔,𝑞³𝜔²𝑓𝑞𝜔²,𝑞³𝜔𝑓⁴𝜔,𝑞𝜔²𝜑𝑞. (41)

Acknowledgments

This work was supported by the National Science Foundation of China, Project nos. 11071107, 11371184, and U1304103. The author would like to thank the referee and editor for many valuable comments and suggestions.