The most significant of the Nørlund means are the Cesàro sums. Here, if we define the sequence pk by then the Cesàro sum Ck is defined by Cesàro sums are Nørlund means if , and hence are regular, linear, stable, and consistent. C0 is ordinary summation, and C1 is ordinary Cesàro summation. Cesàro sums have the property that if then Ch is stronger than Ck. Abelian means Suppose } is a strictly increasing sequence tending towards infinity, and that . Suppose converges for all real numbers x > 0. Then the Abelian mean Aλ is defined as More generally, if the series for f only converges for large x but can be analytically continued to all positive real x, then one can still define the sum of the divergent series by the limit above. A series of this type is known as a generalized Dirichlet series; in applications to physics, this is known as the method of heat-kernel regularization. Abelian means are regular and linear, but not stable and not always consistent between different choices of λ. However, some special cases are very important summation methods. Abel summation If , then we obtain the method of Abel summation. Here where z = exp(−x). Then the limit of f(x) as x approaches 0 through positive reals is the limit of the power series for f(z) as z approaches 1 from below through positive reals, and the Abel sum A(s) is defined as Abel summation is interesting in part because it is consistent with but more powerful than Cesàro summation: whenever the latter is defined. The Abel sum is therefore regular, linear, stable, and consistent with Cesàro summation. Lindelöf summation If , then (indexing from one) we have Then L(s), the Lindelöf sum , is the limit of f(x) as x goes to positive zero. The Lindelöf sum is a powerful method when applied to power series among other applications, summing power series in the Mittag-Leffler star. If g(z) is analytic in a disk around zero, and hence has a Maclaurin series G(z) with a positive radius of convergence, then in the Mittag-Leffler star. Moreover, convergence to g(z) is uniform on compact subsets of the star. Analytic continuation Several summation methods involve taking the value of an analytic continuation of a function. Analytic continuation of power series If Σanxn converges for small complex x and can be analytically continued along some path