# Invariants associated with the epsilon algorithm and its first confluent form

Article

## Summary

It is shown that if numbers εr(m)(r=0, 1, ..., 2n;m=0, 1, ...) can be constructed from the initial values εr(m)=0 (m=1, 2, ...), εr(m)=Sm (m=0, 1, ...) by systematic use of the relationships εr+1(m)r-1(m+1)+(εr(m+D)r(m))–1 (r=0, 1, ..., 2n-1;m=0, 1, …) and an irreducible relationship of the form$$\sum\limits_{v = o}^n {C_v S_{m + v} = G}$$. (m=0, 1, ...) withG=0 holds among the members of the sequence {Sm}, then$$\sum\limits_{r = o}^{2n - 2} {( - 1)^r \varepsilon _r^{(m)} \varepsilon _{r + 1}^{(m)} = - \frac{{\left\{ {\sum\limits_{v = 1}^n {vCv} } \right\}}}{{\left\{ {\sum\limits_{v = 0}^n {Cv} } \right\}}}}$$ (m=0, 1, ...) Furthermore, if functions εr(μ) (r=0, 1, ..., 2n) are constructed from the initial values ε–1(μ)=0, ε0(μ)=S(μ) by systematic use of the relationships$$\varepsilon _{r + 1} (\mu ) = \varepsilon _{r - 1} (\mu ) + \left\{ {\frac{{d\varepsilon _r (\mu )}}{{d\mu }}} \right\}^{ - 1}$$ (r=0, 1, ..., 2n–1) and the functionsS (μ) satisfies an irreducible equation of the form$$\sum\limits_{v = 0}^n {c_v \frac{{d^v S(\mu )}}{{d\mu ^v }} = G}$$ withG=0, then$$\sum\limits_{r = 0}^{2n - 2} {( - 1)^r \varepsilon _r (\mu )\varepsilon _{r + 1} (\mu ) = - \frac{{c_1 }}{{c_0 }}}$$ identically.

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