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A summation method for series of numbers or functions, introduced by F. Hausdorff [[#References|[1]]]; it is defined as follows. A sequence  $  s = \{ s _ {n} \} $
is subjected in succession to three linear matrix transformations:

$$ 
t  =  \delta s,\ \ 
\tau  =  \mu t,\ \ 
\sigma  =  \delta t,
$$

where  $  \delta $
is the transformation by means of the triangular matrix  $  \{ \delta _ {nk} \} $:

$$ 
\delta _ {nk}  = \ 
\left \{

\begin{array}{ll}
(- 1)  ^ {k} \left ( \begin{array}{c}
n \\
 k 
\end{array}
 \right ) ,  & k \leq  n,  \\
0 ,  & k > n;  \\
\end{array}

\right.

$$

and  $  \mu $
is the diagonal transformation by means the diagonal matrix  $  \| \mu _ {nk} \| $:

$$ 
\mu _ {nk}  = \ 
\left \{

\begin{array}{ll}
\mu _ {n} ,  & k = n,  \\
0,  & k \neq n,  \\
\end{array}

 \right .$$

where  $  \mu _ {n} $
is a numerical sequence. The transformation

$$ 
\sigma  =  \lambda s,
$$

where  $  \lambda = \delta \mu \delta $, 
$  \{ \mu _ {n} \} $
is an arbitrary numerical sequence, is called a general Hausdorff transformation, and the matrix  $  \delta \mu \delta $— 
a Hausdorff matrix. Written out, a general Hausdorff transformation has the form

$$ 
\sigma _ {n}  = \ 
\sum _ {k = 0 } ^ { n } 
\lambda _ {nk} s _ {k} ,
$$

where

$$ 
\lambda _ {nk}  = \ 
\left \{

\begin{array}{ll}
\left ( \begin{array}{c}
n \\
 k 
\end{array}
 \right )
\Delta ^ {n - k } \mu _ {k} ,  & k \leq  n,  \\
0,  & k > n;  \\
\end{array}
\right.
$$

$$ 
\Delta \mu _ {k}  =  \mu _ {k} - \mu _ {k + 1 }  ,\  \Delta
 ^ {n} \mu _ {k}  =  \Delta ^ {n - 1 } \mu _ {k} - \Delta ^ {n - 1 } \mu _ {k + 1 }  .
$$

The series

$$ 
\sum _ {n = 0 } ^  \infty   a _ {n}  $$

with partial sums  $  s _ {n} $
is summable by the Hausdorff method to sum  $  S $
if

$$ 
\lim\limits _ {n \rightarrow \infty } \ 
\sigma _ {n}  =  S.
$$

The field and the regularity of the Hausdorff method depend on the sequence  $  \{ \mu _ {n} \} $. 
If  $  \{ \mu _ {n} \} $
is a real sequence, then for the regularity of the method it is necessary and sufficient that  $  \{ \mu _ {n} \} $
is the difference of two absolutely-monotone sequences and that

$$ 
\lim\limits _ {m \rightarrow \infty } \ 
\Delta  ^ {m} \mu _ {m}  =  0; \ \ 
\mu _ {0}  =  1;
$$

or, in another terminology, necessary and sufficient is that the  $  \mu _ {n} $
are regular moments.

The Hausdorff summation method contains as special cases a number of other well-known summation methods. Thus, for  $  \mu _ {n} = 1/( q + 1)  ^ {n} $
the Hausdorff method turns into the Euler method  $  ( E, q) $, 
for  $  \mu = 1/( n + 1)  ^ {k} $
into the Hölder method  $  ( H, k) $, 
and for

$$ 
\mu _ {n}  =  
\frac{1}{\left ( \begin{array}{c}
n + k \\
 k 
\end{array}
 \right ) }

$$

into the Cesàro method  $  ( C, k) $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Hausdorff,   "Summationsmethoden und Momentfolgen I, II"  ''Math. Z.'' , '''9'''  (1921)  pp. 74–109; 280–299</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.H. Hardy,   "Divergent series" , Clarendon Press  (1949)</TD></TR></table>