Invariants associated with the epsilon algorithm and its first confluent form

Summary

It is shown that if numbers ε ^(m)_r (r=0, 1, ..., 2n;m=0, 1, ...) can be constructed from the initial values ε ^(m)_r =0 (m=1, 2, ...), ε ^(m)_r =S _m (m=0, 1, ...) by systematic use of the relationships ε ^(m)_r+1 =ε ^(m+1)_r-1 +(ε ^(m+D)_r -ε ^(m)_r )^–1 (r=0, 1, ..., 2_n-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 {S _m}, 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 ε_r(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.