Whittaker's root series is a series of ratios of determinants for computing a root of an analytic function (Whittaker 1917, pp. 103-106). It applies to polynomials as well as convergent power series.
Let
 |
(1)
|
where 
,
and suppose 
is an analytic function in the disk 
. Suppose 
is a root with 
and every other distinct root
in this disk has absolute
value strictly greater than 
. The multiplicity of
may exceed one.
Set 
for 
,
and also above the polynomial degree when 
is a polynomial.
Define 
and
 |  |  |
(2)
|
 |  |  |
(3)
|
where the matrices defining 
are upper Hessenberg matrices
and Toeplitz matrices. Thus 
, 
, 
, and 
.
If 
for every positive integer 
, then
 |
(4)
|
The partial sums satisfy
 |
(5)
|
The hypothesis on the root of smallest absolute value alone does not ensure that all the denominators
in the series are nonzero.
Even when some early 
vanish, the ratio 
is defined for all sufficiently large 
and tends to 
.
For a short proof, write
 |
(6)
|
where 
.
Equating coefficients, or applying Cramer's
rule, gives 
.
The determinant identity
 |
(7)
|
for 
gives a telescoping series with 
and 
. The unique nearest pole of 
is 
. Its finite principal part
gives coefficients equal to 
times a polynomial in
with polynomial degree one less than the multiplicity of 
, while the remaining coefficients
are exponentially smaller. It follows that 
.
For example, 
gives
 |
(8)
|
