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Harmonic progressions

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The terms of an harmonic progression are the reciprocal of the terms of an arithmetic progression. -terms harmonic progressions are sequences of the form (with for an infinity of terms)

where and are constants. For example, {1/4, 1/16, 1/28, 1/40, 1/52, 1/64, 1/76, 1/88, 1/100, 1/112, ...} (1 / A017569) is an harmonic progression with and . In terms of growth of sequences, nonconstant harmonic progressions have inverse linear growth.

Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms, i.e.

"Primitive" versus "nonprimitive" harmonic progressions

An harmonic progression might be said to be "primitive" if and are coprime. An harmonic progression with (cf. gcd), which might thus be said to be "nonprimitive", is times the corresponding "primitive" harmonic progression. For example {1/4, 1/16, 1/28, 1/40, 1/52, 1/64, 1/76, 1/88, 1/100, 1/112, ...} = (1/4) × {1/1, 1/4, 1/7, 1/10, 1/13, 1/16, 1/19, 1/22, 1/25, 1/28, ...}.

Recurrence

Generating functions

Harmonic progressions have [ordinary] generating functions of the form

where is the generalized hypergeometric function.[1]

See also

Notes

  1. Weisstein, Eric W., Generalized Hypergeometric Function, from MathWorld—A Wolfram Web Resource.