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Double Series

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A double sum is a series having terms depending on two indices,

sum_(i,j)b_(ij).
(1)

A finite double series can be written as a product of series

sum_(i=1)^(m)sum_(j=1)^(n)x_iy_j=x_1y_1+x_1y_2+...+x_2y_1+x_2y_2+...
(2)
=(x_1+x_2+...+x_m)y_1+(x_1+x_2+...+x_m)y_2+...
(3)
=(sum_(i=1)^(m)x_i)(y_1+y_2+...+y_n)
(4)
=(sum_(i=1)^(m)x_i)(sum_(j=1)^(n)y_j).
(5)

An infinite double series can be written in terms of a single series

sum_(k=0)^inftysum_(l=0)^inftyc_(kl)=sum_(i=0)^inftyb_i
(6)

by reordering as follows,

b_0=c_(00)
(7)
b_1=(c_(10)+c_(11))+c_(01)
(8)
b_2=(c_(20)+c_(21)+c_(22))+(c_(02)+c_(12))
(9)
b_3=(c_(30)+c_(31)+c_(32)+c_(33))+(c_(03)+c_(13)+c_(23)).
(10)

Many examples exists of simple double series that cannot be computed analytically, such as the Erdős-Borwein constant

sum_(i=1)^(infty)sum_(j=1)^(infty)1/(2^(ij))=sum_(k=1)^(infty)1/(2^k-1)
(11)
=1-(psi_(1/2)(1))/(ln2)
(12)
=1.60669515...
(13)

(OEIS A065442), where psi_q(z) is a q-polygamma function.

Another series is

sum_(m=1)^(infty)sum_(n=1)^(infty)1/(m^2(m^2+mn+n^2))=1/3sqrt(3)isum_(m=1)^(infty)(H_(-mzeta_2)-H_(mzeta_1))/(m^3)
(14)
=1.004457198...
(15)

(OEIS A091349), where H_n is a harmonic number and zeta_k=e^(2piik/3) is a cube root of unity.

A double series that can be done analytically is given by

zeta(2)=sum_(i=1)^inftysum_(j=1)^infty((i-1)!(j-1)!)/((i+j)!),
(16)

where zeta(2) is the Riemann zeta function zeta(2) (B. Cloitre, pers. comm., Dec. 9, 2004).

The double series

S=sum_(m=1)^inftysum_(n=1)^infty(m^2n)/(3^m(n·3^m+m·3^n))
(17)

can be evaluated by interchanging m and n and averaging,

S=1/2[sum_(m=1)^(infty)sum_(n=1)^(infty)(m^2n)/(3^m(n·3^m+m·3^n))+sum_(m=1)^(infty)sum_(n=1)^(infty)(n^2m)/(3^n(n·3^m+m·3^n))]
(18)
=1/2sum_(m=1)^(infty)sum_(n=1)^(infty)(mn)/(3^(m+n))
(19)
=1/2(sum_(m=1)^(infty)m/(3^m))^2
(20)
=9/(32)
(21)

(Borwein et al. 2004, p. 54).

Identities involving double sums include the following:

sum_(p=0)^inftysum_(q=0)^pa_(q,p-q)=sum_(m=0)^inftysum_(n=0)^inftya_(n,m)=sum_(r=0)^inftysum_(s=0)^(|_r/2_|)a_(s,r-2s),
(22)

where

|_r/2_|={1/2r r even; 1/2(r-1) r odd
(23)

is the floor function, and

sum_(i=1)^nsum_(j=1)^nx_ix_j=n^2<x^2>.
(24)

Consider the series

S(a,b,c;s)=sum^'_(m,n=-infty)^infty(am^2+bmn+cn^2)^(-s)
(25)

over binary quadratic forms, where the prime indicates that summation occurs over all pairs of m and n but excludes the term (m,n)=(0,0). If S can be decomposed into a linear sum of products of Dirichlet L-series, it is said to be solvable. The related sums

S_1(a,b,c;s)=sum^'_(m,n=-infty)^infty(-1)^m(am^2+bmn+cn^2)^(-s)
(26)
S_2(a,b,c;s)=sum^'_(m,n=-infty)^infty(-1)^n(am^2+bmn+cn^2)^(-s)
(27)
S_(1,2)(a,b,c;s)=sum^'_(m,n=-infty)^infty(-1)^(m+n)(am^2+bmn+cn^2)^(-s)
(28)

can also be defined, which gives rise to such impressive formulas as

S_1(1,0,58;1)=-(piln(27+5sqrt(29)))/(sqrt(58))
(29)

(Glasser and Zucker 1980). A complete table of the principal solutions of all solvable S(a,b,c;s) is given in Glasser and Zucker (1980, pp. 126-131).

The lattice sum b_2(2s) can be separated into two pieces,

b_2(2s)=sum^'_(i,j=-infty)^infty((-1)^(i+j))/((i^2+j^2)^s)
(30)
=sum_(i=1)^(infty)sum_(j=1)^(infty)((-1)^(i+j))/((i^2+j^2)^s)+sum_(i=1)^(infty)sum_(j=-1)^(-infty)((-1)^(i+j))/((i^2+j^2)^s)+sum_(i=-1)^(-infty)sum_(j=1)^(infty)((-1)^(i+j))/((i^2+j^2)^s)+sum_(i=-1)^(-infty)sum_(j=-1)^(-infty)((-1)^(i+j))/((i^2+j^2)^s)+sum_(j=-infty)^(-1)((-1)^j)/(j^(2s))+sum_(j=1)^(infty)((-1)^j)/(j^(2s))+sum_(i=-infty)^(-1)((-1)^i)/(i^(2s))+sum_(i=1)^(infty)((-1)^i)/(i^(2s))
(31)
=4[sum_(i,j=1)^(infty)((-1)^(i+j))/((i^2+j^2)^s)+sum_(i=1)^(infty)((-1)^i)/(i^(2s))]
(32)
=4[sum_(i,j=1)^(infty)((-1)^(i+j))/((i^2+j^2)^s)-eta(2s)],
(33)

where eta(n) is the Dirichlet eta function. Using the analytic form of the lattice sum

b_2(s)=-4beta(s)eta(s)
(34)
=4[S_(1,2)(1,0,1;s)-eta(2s)],
(35)

where beta(s) is the Dirichlet beta function gives the sum

S_(1,2)(1,0,1;s)=sum_(i,j=1)^(infty)((-1)^(i+j))/((i^2+j^2)^s)
(36)
=eta(2s)-eta(s)beta(s).
(37)

Borwein and Borwein (1987, p. 291) show that for R[s]>1,

sum^'_(i,j=-infty)^infty1/((i^2+j^2)^s)=4beta(s)zeta(s)
(38)
sum^'_(i,j=-infty)^infty((-1)^j)/((i^2+j^2)^s)=2^(-s)b_2(2s),
(39)

where zeta(s) is the Riemann zeta function, and for appropriate s,

sum_(i,j=1)^(infty)((-1)^(i+j))/((i+j)^s)=eta(s)-eta(s-1)
(40)
sum_(i,j=1)^(infty)((-1)^(i+1))/((i+j)^s)=2^(-s)zeta(s)
(41)
sum_(i,j=1)^(infty)1/((i+j)^s)=zeta(s-1)-zeta(s)
(42)
sum^'_(i,j=-infty)^infty((-1)^(i+j+1))/((|i|+|j|)^s)=4eta(s-1)
(43)
sum^'_(i,j=-infty)^infty1/((i+j)^s)=4zeta(s-1)
(44)
sum_(i,j=0)^(infty)((-1)^(i+j))/((2i+j+1)^s)=1/2(1-2^(-s))eta(s)+1/2beta(s)
(45)

(Borwein and Borwein 1987, p. 305).

Another double series reduction is given by

sum_(m,n=-infty)^infty(F(|2m+2n+1|))/(cosh[(2n+1)u]cosh(2nu))=2sum_(n=0)^infty((2n+1)F(2n+1))/(sinh[(2n+1)u]),
(46)

where F denotes any function (Glasser 1974).

See also

Euler Sum, Lattice Sum, Madelung Constants, Multiple Series, Multivariate Zeta Function, Series, Triple Series, Weierstrass's Double Series Theorem

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References

Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.Glasser, M. L. "Reduction Formulas for Multiple Series." Math. Comput. 28, 265-266, 1974.Glasser, M. L. and Zucker, I. J. "Lattice Sums." In Perspectives in Theoretical Chemistry: Advances and Perspectives, Vol. 5 (Ed. H. Eyring). New York: Academic Press, pp. 67-139, 1980.Hardy, G. H. "On the Convergence of Certain Multiple Series." Proc. London Math. Soc. 2, 24-28, 1904.Hardy, G. H. "On the Convergence of Certain Multiple Series." Proc. Cambridge Math. Soc. 19, 86-95, 1917.Jeffreys, H. and Jeffreys, B. S. "Double Series." §1.053 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 16-17, 1988.Meyer, B. "On the Convergence of Alternating Double Series." Amer. Math. Monthly 60, 402-404, 1953.Móricz, F. "Some Remarks on the Notion of Regular Convergence of Multiple Series." Acta Math. Hungar. 41, 161-168, 1983.Sloane, N. J. A. Sequences A065442 and A091349 in "The On-Line Encyclopedia of Integer Sequences."Wilansky, A. "On the Convergence of Double Series." Bull. Amer. Math. Soc. 53, 793-799, 1947.Zucker, I. J. and Robertson, M. M. "Some Properties of Dirichlet L-Series." J. Phys. A: Math. Gen. 9, 1207-1214, 1976a.Zucker, I. J. and Robertson, M. M. "A Systematic Approach to the Evaluation of sum_((m,n!=0,0))(am^2+bmn+cn^2)^(-s)." J. Phys. A: Math. Gen. 9, 1215-1225, 1976b.

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Double Series

Cite this as:

Weisstein, Eric W. "Double Series." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DoubleSeries.html

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