Abstract
Formulas are proved for the number of representations of a positive integer by each of the four quaternary quadratic forms x2+y2+2z2+3t2, x2+2y2+2z2+6t2, x2+3y2+3z2+6t2 and 2x2+3y2+6z2+6t2. As a consequence of these formulas, each of the four series [Formula: see text] is determined in terms of Ramanujan's theta function.
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