Journl of Alger umer Teor: Avne n Applon Volume umer 9 Pge -7 MATRICES WITH COVOLUTIOS OF BIOMIAL FUCTIOS THEIR DETERMIATS AD SOME EXAMPLES ORMA C SEVERO n PAUL J SCHILLO Meove Lne Wllmvlle Y USA e-ml: evero@uffloeu 88 M Sree Bufflo Y USA Ar Te purpoe of rle o on e eermnn of n memer of l of mre oe enre re onvoluon of noml funon We lo gve ere ome emple n e form of prolem n oluon ue ome of e propere of ee mre Inrouon We u mre efne n [] ve elemen re onvoluon of noml erm Te of e pper ell-non enque from nvrn eor e mmer repreenon of mre e e repreenon of e mulplve emgroup enor Te generl e for mre of n e ell-non (ee [] []) Mem Suje Clfon: A A99 Keor n pre: mre onvoluon of noml funon eermnn Krou mr Reeve Jnur 9; Reve Aprl 9 9 Senf Avne Puler
ORMA C SEVERO n PAUL J SCHILLO For omple numer n e enoe M e mr oe - ro n - olumn enr K ( ) ; λ () Te mr M equl for e ornr mr for n for () llure noer meo for generng e mr (ee[]) (In e Appen e pl e mre for n ) In [] e follong mulplon eorem prove for K n for n omple numer n : () To full ppree e poer of omomorpm e enourge e reer o verf () for le n n for ome enre for n Aloug e eermnn lulon of n mr n e one rng e mr n upper-rngulr form (ee eg []) e erve e reul rel ung generng funon of e mre Te Deermnn of M In e follong eorem e ue n lernve efnon of M lo gven n [] Te - olumn K of M -
MATRICES WITH COVOLUTIOS OF BIOMIAL elemen λ ( ; ) e oeffen of u for K n e generng funon G ( ; u) ( u) ( u) Teorem Te eermnn of M for K ( ) / D ( ) () Proof We efne E e eermnn olumn equl o of D n - olumn generng funon K equl o G ( ; u) G ( ; u) ( / ) ( u ) ( u) ( u) ( ) ( u) ( / ) ( u) ( u) [( u) ( u) / ] G ( ; u) u ere Tu D equl o e eermnn E e olumn e me of D n - olumn K vng enre for n λ ( ; ) for K o ung Lple formul o epn E long e ro e ge D C( ) D ere C ( ) e eermnn of e gonl mr e of oe enre Fnll ung e nuon poe D / ( ) ( ) rvll rue for e ge
ORMA C SEVERO n PAUL J SCHILLO ( ) ( ) ( ) ( ) / / D T omplee e proof oe for emple e eermnn of M gven n e Appen mpl ( ) Alo from [] e Krou mr p B o eermnn ( ) / p B T mple reul pper o e ne Some Iene Te follong ene re erve ung eer Equon () () or () Prolem For non-negve neger n n omple numer n o ( ) oe Ue e equon n e f e enr n e - ro n - olumn of e mr
MATRICES WITH COVOLUTIOS OF BIOMIAL Prolem From e relon n e mgnr un n non-negve neger o for K ( ) ( ) Prolem For non-negve neger n n rel numer n le A n B e e mre oe - ro n - olumn enre re repevel ( ) ( ) n ( ) ( ) So e eermnn of A n B re e equl o oe A Furermore f H en H A H B Prolem Le D e e eermnn oe - ro - olumn elemen
ORMA C SEVERO n PAUL J SCHILLO ere e mgnr un So e vlue of D ( ) ( ) 8 7 n 8 7 o π π oe D e eermnn of e mr Appen For n e mre M re repevel n
MATRICES WITH COVOLUTIOS OF BIOMIAL 7 Referene [] J A Deuonné n J B Crrell Invrn Teor Ol n e Aem Pre e Yor n Lonon (97) MR 79 [] O Hol Evluon of Slveer pe eermnn ung lo-rngulron Proeeng of e - ISAAC Congre (H G W Beger e l e) Worl Senf () 9- (eprn rxv:m/) [] C Severo Mre onvoluon of noml funon n Krou mre Lner Alger Appl 9 (8) - [] T A Sprnger Invrn Teor Leure oe n M 8 Sprnger-Verlg Berln n e Yor (977) MR78 g