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Borel-Regularized Sum


The great success mathematicians had studying hypergeometric functions _pF_q(a_1,...,a_p;b_1,...,b_q;z) for the convergent cases (p<=q+1) prompted attempts to provide interpretations for such functions in divergent cases when p>q+1. An interesting approach to interpreting sums of these divergent series was suggested by Emile Borel in 1899. By definition the generalized Borel sum of an arbitrary series

 phi(z)=sum_(k=0)^inftyc_kz^k
(1)

is the value of the integral

 B(z)=int_0^infty...int_0^inftye^(-t_1-t_2-...-t_r)(sum_(k=0)^infty(c_kz^k)/((k!)^r)t_1^kt_2^k...t_r^k)dt_1...dt_r
(2)

with r in Z^+.

This definition allows interpretation of the sums of divergent hypergeometric series as generalized Borel sums, where these Borel sums always coincide with other convergent hypergeometric series.

Consider an example related to the asymptotic formula for the function _2F_0(a,b;;z), beginning with the relation

 _2F_0(a,b;;z)=(-1/z)^aU(a,1+a-b,-1/z).
(3)

The right-hand side of the above formula can be interpreted as a Borel sum (with r=1) of the classically divergent series _2F_0(a,b;;z). Choose c_k=(a)_k(b)_k/k! so that

 phi(z)=sum_(k=0)^infty((a)_k(b)_k)/(k!)z^k=_2F_0(a,b;;z).
(4)

Then

 sum_(k=0)^infty(c_kz^k)/(k!)t^k=sum_(k=0)^infty((a)_k(b)_k(zt)^k)/(k!^2)=_2F_1(a,b;1;zt),
(5)

so the Borel-regularized sum B(z) for the divergent series _2F_0(a,b;;z) is equal to

B(z)=int_0^inftye^(-t)_2F_1(a,b;1;zt)dt
(6)
=(-1/z)^aU(a,1+a-b,-1/z).
(7)

Borel's approach to the summation of divergent series was not investigated deeply for hypergeometric series, however; the most effective results for hypergeometric series were found with the Mellin-Barnes integral later on.


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Cite this as:

Weisstein, Eric W. "Borel-Regularized Sum." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Borel-RegularizedSum.html

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