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A method for summing series of numbers and functions; denoted by $(R,\lambda,k)$. A series $\sum_{n=0}^\infty a_n$ is summable by the Riesz summation method $(R,\lambda,k)$ to the sum $s$ if

$$\lim_{\omega\to+\infty}\sum_{\lambda_n\leq\omega}\left(1-\frac{\lambda_n}{\omega}\right)^ka_n=s,$$

where $k>0$, $0\leq\lambda_0<\ldots<\lambda_n\to\infty$, and $\omega$ is a continuous parameter. The method was introduced by M. Riesz [[#References|[1]]] for the summation of [[Dirichlet series|Dirichlet series]]. The method $(R,\lambda,k)$ is regular; when $\lambda_n=n$ it is equivalent to the Cesàro summation method $(C,k)$ (cf. [[Cesàro summation methods|Cesàro summation methods]]), and these methods are compatible (cf. [[Compatibility of summation methods|Compatibility of summation methods]]).

Riesz considered also a method in which summability of the series $\sum_{n=0}^\infty a_n$ is defined by means of the limit of the sequence $\{\sigma_m\}$, where

$$\sigma_m=\frac{1}{P_m}\sum_{k=0}^mp_ks_k,$$

$$P_m=\sum_{k=0}^mp_k\neq0,\quad s_k=\sum_{n=0}^ka_n.$$

This method is denoted by $(R,p_n)$. The method $(R,\lambda,k)$ is a modification of the method $(R,p_n)$ (when $k=1$) and is a generalization of it to the case of an arbitrary $k>0$.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Riesz,   "Une méthode de sommation équivalente à la méthode des moyennes arithmétique"  ''C.R. Acad. Sci. Paris'' , '''152'''  (1911)  pp. 1651–1654</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Riesz,   "Sur la sommation des séries de Dirichlet"  ''C.R. Acad. Sci. Paris'' , '''149'''  (1909)  pp. 18–21</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.H. Hardy,   M. Riesz,   "The general theory of Dirichlet series" , Cambridge Univ. Press  (1915)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G.H. Hardy,   "Divergent series" , Clarendon Press  (1949)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Zeller,   W. Beekmann,   "Theorie der Limitierungsverfahren" , Springer  (1970)</TD></TR></table>