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Research article

Generalised Kähler structure on CP2 and elliptic functions

  • Received: 21 December 2021 Revised: 01 September 2022 Accepted: 04 October 2022 Published: 02 February 2023
  • Primary: 53D18; Secondary: 53D17, 53C15

  • We construct a toric generalised Kähler structure on CP2 and show that the various structures such as the complex structure, metric etc are expressed in terms of certain elliptic functions. We also compute the generalised Kähler potential in terms of integrals of elliptic functions.

    Citation: Francesco Bonechi, Jian Qiu, Marco Tarlini. Generalised Kähler structure on CP2 and elliptic functions[J]. Journal of Geometric Mechanics, 2023, 15(1): 188-223. doi: 10.3934/jgm.2023009

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  • We construct a toric generalised Kähler structure on CP2 and show that the various structures such as the complex structure, metric etc are expressed in terms of certain elliptic functions. We also compute the generalised Kähler potential in terms of integrals of elliptic functions.



    In [18] Ramanujan showed a total of 17 series for 1/π but he did not indicate how he arrived at these series. The Borwein brothers [5] gave rigorous proofs of Ramanujan's series for the first time and also obtained many new series for 1/π. Till now, many new Ramanujan's-type series for 1/π have been published, (see, for example, [4,6,8]). Chu [7], Liu [15,16] and Wei et al. [21,22] gave many π-formula with free parameters by means of gamma functions and hypergeometric series. Guillera [10] proved a kind of bilateral semi-terminating series related to Ramanujan-like series for negative powers of π. Moreover, Guillera and Zudilin [11] outlined an elementary method for proving numerical hypergeometric identities, in particular, Ramanujan-type identities for 1/π. Recently, q-analogues of Ramanujan-type series for 1/π have caught the interests of many authors (see, for example, [9,12,13,14,20,21]).

    Although various definitions for gamma functions are used in the literature, we adopt the following definition [23, p.76]

    1Γ(z)=zeγzn=1(1+zn)ezn

    where γ is the Euler constant defined as

    γ=limn(1+12++1nlogn).

    It is easy to verify that Γ(1)=1,Γ(12)=π and Γ(z+1)=zΓ(z). It follows that for every positive integer n, Γ(n)=(n1)!.

    For any complex α, we define the general rising shifted factorial by

    (z)α=Γ(z+α)/Γ(z). (1.1)

    Obviously, (z)0=1. For every positive integer n, we have

    (z)n=Γ(z+n)/Γ(z)=z(z+1)(z+n1)

    and

    (z)n=Γ(zn)/Γ(z)=1(z1)(z2)(zn).

    For convenience, we use the following compact notations

    (a1,a2,,am)n=(a1)n(a2)n(am)n

    and

    (a)(n1,n2,,nm)=(a)n1(a)n2(a)nm.

    Following [1,3], the hypergeometric series is defined by

    r+1Fs[a0,a1,,arb1,,bs;z]=k=0(a0,a1,,ar)k(b1,,bs)kzkk!,

    where ai,bj(i=0,1,,r,j=1,2,,s) are complex numbers such that no zero factors appear in the denominators of the summand on the right hand side.

    We let Fp:r;uq:s;v (p,q,r,s,u,vN0={0,1,2,}) denote a general (Kampé de Fériet's) double hypergeometric function defined by (see [2,19])

    Fp:r;uq:s;v[α1,,αp:a1,,ar;c1,,cu;β1,,βq:b1,,bs;d1,,dv;x,y]=m,n=0(α1,,αp)m+n(a1,,ar)m(c1,,cu)n(β1,,βq)m+n(b1,,bs)m(d1,,dv)nxmm!ynn!,

    where, for convergence of the double hypergeometric series,

    p+rq+s+1andp+uq+v+1,

    with equality only when |x| and |y| are appropriately constrained (see, for details, [19,Eq 1.3(29),p.27]).

    There exist numerous identities for such series. For example, we have

    Theorem 1.1 (See [17,(30)] ) If Re(ed)>0 and Re(d+eabc)>0, then

    F0:3;31:1;1[:a,b,c;da,db,dc;d:e;d+eabc;1,1]=Γ(e)Γ(e+dabc)Γ(ed)Γ(ea)Γ(eb)Γ(ec).

    In [15], Liu used the general rising shifted factorial and the Gauss summation formula to prove the following four-parameter series expansions formula, which implies infinitely many Ramanujan type series for 1/π and π.

    Theorem 1.2 For any complex α and Re(cab)>0, we have

    n=0(α)a+n(1α)b+nn!Γ(c+n+1)=(α)a(1α)bΓ(cab)(α)cb(1α)casinπαπ.

    Motivated by Liu's work, in this paper we derive the following result from Theorem 1.1 which enables us to give many double series expansions for 1/π and  π. To the best of our knowledge, most of the results in this paper have not previously appeared.

    Theorem 1.3 If dN0,Re(ed+σδ)>0 and Re(d+eabc+δ+σαβγ)>0, then

    m,n=0(α)a+m(β)b+m(γ)c+m(δα)da+n(δβ)db+n(δγ)dc+nm!n!(δ+d)m+n(σ)e+m(δ+σαβγ)d+eabc+n=(α)a(β)b(γ)c(δα)da(δβ)db(δγ)dc(σδ)ed(σα)ea(σβ)eb(σγ)ecΓ(σ)Γ(σδ)Γ(δ+σαβγ)Γ(σα)Γ(σβ)Γ(σγ).

    Several examples of such formulae are

    m,n=0(12)3m(12)2nm!n!(m+n)!(m+1)!(2n+1)=4π,
    m,n=0(12)3m(32)3nm!n!(m+n)!(n+3)!(12)m+1=π,

    and

    m,n=0(23)2m(13)3nm!n!(n+1)!(23m)(13)m+n=3π3.

    The remainder of the paper is organized as follows. In section 2 we give the proof of Theorem 1.3. Sections 3 and 4 are devoted to the double series expansions for 1/π and π, respectively.

    First of all, by making use of (1.1), Theorem 1.3 can be restated as follows:

    m,n=0Γ(a+m)Γ(b+m)Γ(c+m)Γ(da+n)Γ(db+n)Γ(dc+n)m!n!Γ(d+m+n)Γ(e+m)Γ(d+eabc+n)=Γ(a)Γ(b)Γ(c)Γ(da)Γ(db)Γ(dc)Γ(ed)Γ(d)Γ(ea)Γ(eb)Γ(ec). (2.1)

    From (1.1) it is easy to see that

    Γ(a+α+m)=(α)a+mΓ(α), Γ(b+β+m)=(β)b+mΓ(β), Γ(c+γ+m)=(γ)c+mΓ(γ),Γ(da+δα+n)=(δα)da+nΓ(δα), Γ(db+δβ+n)=(δβ)db+nΓ(δβ),Γ(dc+δγ+n)=(δγ)dc+nΓ(δγ), Γ(d+δ+m+n)=(δ)d+m+nΓ(δ)Γ(e+m+σ)=(σ)e+mΓ(σ), Γ(a+α)=(α)aΓ(α), Γ(b+β)=(β)bΓ(β), Γ(c+γ)=(γ)cΓ(γ),Γ(da+δα)=(δα)daΓ(δα), Γ(db+δβ)=(δβ)dbΓ(δβ),Γ(dc+δγ)=(δγ)dcΓ(δγ), Γ(ed+σδ)=(σδ)edΓ(σδ),Γ(d+δ)=(δ)dΓ(δ),Γ(ea+σα)=(σα)eaΓ(σα),Γ(eb+σβ)=(σβ)ebΓ(σβ), Γ(ec+σγ)=(σγ)ecΓ(σγ),Γ(d+eabc+δ+σαβγ)=(δ+σαβγ)d+eabcΓ(δ+σαβγ).

    and we realize that (δ)d+m+n=(δ)d(δ+d)m+n when dN0. Replacing(a,b,c,d,e) by (a+α,b+β,c+γ,d+δ,e+σ) in (2.1) and substituting above identities into the resulting equation, we get the desired result.

    In this section we will use Theorem 1.3 to prove the following double series expansion formula for 1/π.

    Theorem 3.1 If dN0,Re(ed+1)>0 and Re(d+eabc+32)>0, then

    m,n=0(12)(a+m,b+m,c+m,da+n,db+n,dc+n)m!n!(d+1)m+n(2)e+m(32)d+eabc+n=(12)(a,b,c,da,db,dc)(1)ed(32)(ea,eb,ec)4π.

    Proof. Let (α,β,γ,δ,σ)=(12,12,12,1,2) in Theorem 1.3. We find that

    m,n=0(12)(a+m,b+m,c+m,da+n,db+n,dc+n)m!n!(d+1)m+n(2)e+m(32)d+eabc+n=(12)(a,b,c,da,db,dc)(1)ed(32)(ea,eb,ec)Γ(2)Γ(1)Γ(32)Γ3(32). (3.1)

    Substituting Γ(32)=π2 into (3.1) we obtain the result immediately. Putting (a,b,c)=(0,0,0) in Theorem 3.1 we get the following general double summation formula for 1/π with two free parameters.

    Corollary 3.2 If dN0,Re(ed+1)>0 and Re(d+e+32)>0, then

    m,n=0(12)3(m,d+n)m!n!(d+1)m+n(2)e+m(32)d+e+n=4(12)3d(1)edπ(32)3e.

    Setting d=0 and e=kN0 in Corallary 3.2 we have the following result.

    Proposition 3.3 Let k be a nonnegative integer. Then

    m,n=0(12)3(m,n)m!n!(m+n)!(m+k+1)!(32+k)n=4k!π(32)2k.

    Example 3.1 (k=0 in Proposition 3.3).

    m,n=0(12)3m(12)2nm!n!(m+n)!(m+1)!(2n+1)=4π.

    If d=e=kN0 in Corollary 3.2 we achieve

    Proposition 3.4 Let k be a nonnegative integer. Then

    m,n=0(12)3(m,n+k)m!n!(k+1)m+n(m+k+1)!(32)n+2k=4π(2k+1)3.

    If we put k=0 into Proposition 3.4, then we can also get Example 3.1.

    In this section we will prove the following theorem, which allows us to derive infinitely double series expansions for π.

    Theorem 4.1 If dN0,Re(edσ+1)>0 and Re(d+eabc+2)>0, then

    m,n=0(σ1)(a+m,b+m,c+m)(σ)(da+n,db+n,dc+n)m!n!(2σ+d1)m+n(σ)e+m(2)d+eabc+n=(σ1)(a,b,c)(σ)(da,db,dc)(1σ)ed(1)(ea,eb,ec)πsinσπ.

    Proof. Let (α,β,γ,δ)=(σ1,σ1,σ1,2σ1) in Theorem 1.3. We obtain that

    m,n=0(σ1)(a+m,b+m,c+m)(σ)(da+n,db+n,dc+n)m!n!(2σ+d1)m+n(σ)e+m(2)d+eabc+n=(σ1)(a,b,c)(σ)(da,db,dc)(1σ)ed(1)(ea,eb,ec)Γ(σ)Γ(1σ)Γ(2)Γ3(1). (4.1)

    Combining Γ(σ)Γ(1σ)=πsinσπ with (4.1) we get the desired result immediately. Putting a=b=c=0 in Theorem 4.1 we obtain the following equation.

    Corollary 4.2 If dN0,Re(edσ+1)>0 and Re(d+e+2)>0, then

    m,n=0(σ1)3m(σ)3d+nm!n!(2σ+d1)m+n(σ)e+m(2)d+e+n=(σ)3d(1σ)ed(1)3eπsinσπ.

    Letting σ=12 in Corollary 4.2, we get the following proposition.

    Proposition 4.3 If dN0,Re(ed+12)>0 and Re(d+e+2)>0, then

    m,n=0(12)3m(12)3d+nm!n!(d)m+n(12)e+m(2)d+e+n=(12)3d(12)ed(1)3eπ.

    When we set d=1 and e=kN={1,2,3} in Proposition 4.3 we obtain

    Proposition 4.4 If k is a positive integer, then

    m,n=0(12)3m(32)3nm!n!(m+n)!(n+k+2)!(12)m+k=π(12)k1(k!)3.

    Example 4.1 (k=1 in Proposition 4.4).

    m,n=0(12)3m(32)3nm!n!(m+n)!(n+3)!(12)m+1=π.

    Putting σ=13 in Corollary 4.2, we get the following proposition.

    Proposition 4.5 If dN0,Re(ed+23)>0 and Re(d+e+2)>0, then

    m,n=0(23)3m(13)3d+nm!n!(d13)m+n(13)e+m(2)d+e+n=23π(13)3d(23)ed3(1)3e.

    When we set d=0 and e=kN0 in Proposition 4.5 we obtain

    Proposition 4.6 If k is a nonnegative integer, then

    m,n=0(23)3m(13)3nm!n!(13)m+n(13)m+k(n+k+1)!=23π(23)k3k!3.

    Example 4.2 (k=0 in Proposition 4.6).

    m,n=0(23)2m(13)3nm!n!(n+1)!(23m)(13)m+n=3π3.

    Setting d=e=kN0 in Proposition 4.5, we get

    Proposition 4.7 If k is a nonnegative integer, then

    m,n=0(23)3m(13+k)3nm!n!(n+2k+1)!(k13)m+n(13)m+k=23π3k!3.

    Therefore, Example 4.2 can also be deduced by fixing k=0 in the above equation.

    Example 4.3 (k=1 in Proposition 4.7).

    m,n=0(23)3m(43)3nm!n!(n+3)!(23)m+n(43)m=23π9.

    Double series expansions for 1/π and π with several free parameters are established and many interesting formulas are obtained. A point that should be stressed is that there is an important connection between the summation formulas for double hypergeometric functions and double series expansions for the powers of π.

    The author was partially supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (grant 19KJB110006).

    The author declares that there is no conflict of interest in this paper.



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