Approximation by Spline Functions - download pdf or read online

By Gunther Nurnberger

ISBN-10: 3540516182

ISBN-13: 9783540516187

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Stacks and alternating parity in partitions, Adv. in Math. 5 3 (1984), 55-74. 36 FUNDAMENTAL PROPERTIES OF BASIC HYPERGEOMETRIC SERIES 14. , q-Series: Their development and application in analysis, number theory, combinatorics, physics and computer algebra, CBMS Regional Conf. Ser. , no. 66, Amer. Math. , 1986. 15. G. E. Andrews and R. Askey, Enumeration of partitions: The role of Eulerian series and q-orthogonal polynomials, Higher Combinatorics, M. Aigner, editor, Reidel, Dordrecht, Holland, 1977.

5) Q0(n) + Qxin) = Q\(n) + Q%{n). 2) is clearly the generating function for Qo(n)—Qi{n). Examination will reveal that the second series generates (—l)n{Q\{n) — Q%(ri)). 6) Qo(n) - Q^n) = ( - l ) n ( Q I ( n ) - Q*3(n)). 2) we obtain this common value explicitly: { + 1 if n = 3*i±*, k > 0, - 1 if n = 3*^=*, k > 0, 0 otherwise. 6) that QJ(2n) = Qo(2n); Q*(2n) = Q 1 (2n), QJ(2n + l ) = Q i ( 2 n + l); Q^(2n + 1) = Q 0 (2n + 1). 8) were stated in [16]. Afterwards, D. H. 8). 72). 9) q3t2 q5t3 3Q . )(!

K - 1). With these kp arguments we find, as above Hence Qm — 0 if m ^ 0 (mod &), Now the function Qn is related to the TV-fold application of T~l. 31) 31 g(t)(t)n/(ctq)n. 22). 23) and replacing o by a 1 / * ? - 1 , 0\ 321 V ( Q ' ^ / P ) " p ( C g ) " , n _ (Q;g f c / P )oo(c^)oo y - {t)nkan Now replace qk/p by 0 M" n {t)nkan (a;<7i)oo(c^)oo y > Woo ^ (gi;gi)n(cty)nfc This identity in ^ and q\, as it stands, has meaning only for q\ = qk/p. However, if we use {b;q)oc {b',q)n = (bqn;q)oc it may be written as (21-34) (cq;q)oo v {qn+1\q)ootn te9)oo ~*n (a^ p ;^i)oo(c(7 n+1 ;9)co " 1 y^(^+1;9i)oc(^fen+1;g)ooan (qi;qi)oc^Q (tqkn;q)oo Further, let y — q\ = qk.

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Approximation by Spline Functions by Gunther Nurnberger

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