LAUSR

Abstract The aim of this paper is the study of a transformation dealing with the general K-fold infinite series of the form ∑ n 1 ≥ ⋯ ≥ n K ≥ 1 ∏ j = 1 K a n j , especially those, where a n = R ( n ) is a rational function satisfying certain simple conditions. These sums represent the direct generalization of the well-known multiple Riemann zeta-star function with repeated arguments ζ ⋆ ( { s } K ) when a n = 1 / n s . Our result reduces ∑ ∏ a n j to a special kind of one-fold infinite series. We apply the main theorem to the rational function R ( n ) = 1 / ( ( n + a ) s + b s ) in case of which the resulting K-fold sum is called the generalized multiple Hurwitz zeta-star function ζ ⋆ ( a , b ; { s } K ) . We construct an effective algorithm enabling the complete evaluation of ζ ⋆ ( a , b ; { 2 s } K ) with a ∈ { 0 , − 1 / 2 } , b ∈ R ∖ { 0 } , ( K , s ) ∈ N 2 , by means of a differential operator and present a simple ‘Mathematica’ code that allows their symbolic calculation. We also provide a new transformation of the ordinary multiple Riemann zeta-star values ζ ⋆ ( { 2 s } K ) and ζ ⋆ ( { 3 } K ) corresponding to a = b = 0 .