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DEMONSTRATIO MATHEMATICAVol. XLIINo 22009Alexander E. PatkowskiO N SOME P A R T I T I O N S W H E R E E V E N PARTS D ONOT REPEATA b s t r a c t . We offer some new results on some partition functions in which evenparts do not repeat. In particular, we show certain partition functions in this categoryare lacunary.1. Introduction and Main ResultsLet l(n,m) denote the number of partitions of n in which even partsdo not repeat with m even parts. It is well-known that l(n,m) has thegenerating function(22\W . 9 )ocoooom=0 n=0where we have employed standard notation [9](a-,q)n := (1 - a ) ( l - aq) • • • (1 - a g " " 1 ) ,ool i m ( a ; q)n = (a; q)^:= IT (1 -aq11).n=0The partition function l(n,m) has appeared in numerous studies of wellknown ^-series identities: a (/-series identity due to Lebesgue [3], Andrews'[2] study of some identities of Gauss, and many others. One such (/-seriesidentity is given by(2)n(n+1)/2which, by taking z = — 1 in (1), can be paraphrased to give the interestingpartition theorem [1]:2000 Mathematics Subject Classification: 11B65, 11B75, 11P99.Key words and phrases:
Demonstratio Mathematica – de Gruyter
Published: Apr 1, 2009
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