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An analogue of Kummer congruences for \(q\)-Euler numbers. (Un analogue des congruences de Kummer pour les \(q\)-nombres d’Euler.) (French) Zbl 0485.05006
Es werden \(q\)-Euler-Polynome \(E_{n,n}(q)\) studiert, wobei \(E_{n,n}(1)=E_{n+n}\) die üblichen Euler-Zahlen sind. Das Hauptresultat stellt ein \(q\)-Analogon der Kummerschen Kongruenzen für Euler-Zahlen dar. Ausserdem wird eine kombinatorische Interpretation der \(E_{n,n}(q)\) gegeben.
Reviewer: J. Cigler

11B68 Bernoulli and Euler numbers and polynomials
05A15 Exact enumeration problems, generating functions
Full Text: DOI
[1] Andrews, G.E.; Gessel, I., Divisibility properties of the q-tangent numbers, Proc. amer. math. soc., 68, 380-384, (1978) · Zbl 0401.10020
[2] Andrews, G.E.; Foata, D., Congruences for the q-secant numbers, Eur. J. combin., 1, 283-287, (1981) · Zbl 0455.10006
[3] Askey, R., orthogonal polynomials and special functions, regional conference series in applied math. no. 21, (1975), SIAM Philadelphia
[4] Carlitz, L., q-Bernoulli numbers and polynomials, Duke math. J., 15, 987-1000, (1948) · Zbl 0032.00304
[5] Chu Shih-Chieh, Ssu Yuan Yü Chien, 1303 (en chinois), cf. [3].
[6] Foata, D., Further divisibility properties of the q-tangent numbers, Proc. amer. math. soc., 81, 143-148, (1981) · Zbl 0485.05007
[7] Foata, D.; Schützenberger, M.-P., Major index and inversion number of permutations, Math. nachr., 83, 143-159, (1978) · Zbl 0319.05002
[8] Gessel, I., Generating functions and enumeration of sequences, Doctoral thesis, MIT, (1977)
[9] Jackson, F.H., A basic sine and cosine with symbolical solutions of certain differential equations, Proc. Edinburgh math. soc., 22, 28-39, (1904) · JFM 35.0445.01
[10] Knuth, D.E.; Buckholtz, T.J., Computation of tangent, Euler, and Bernoulli numbers, Math. comp., 21, 663-688, (1967) · Zbl 0178.04401
[11] Kummer, E.F., Über eine allgemeine eigenschaft der rationale entwickelungscoëfficiente einer bestimmten gattung analytischer functionen, J. reine angew. math., 41, 368-372, (1850)
[12] Lucas, E., Sur LES congruences des nombres eulériens et des coefficients différentiels des fonctions trigonométriques, suivant un module premier, Bull. soc. math. France, 6, 49-54, (1878) · JFM 10.0139.04
[13] Nielsen, N., Traité élémentaire des nombres de Bernoulli, (1923), Gauthier-Villars Paris
[14] Rawlings, D.P., Generated worpitzky identities with applications to permutation enumeration, Eur. J. combin., 2, 67-78, (1981) · Zbl 0471.05006
[15] Stanley, R.P., Binomial posets, Möbius inversion and permutation enumeration, J. combin. theory ser. A, 20, 336-356, (1976) · Zbl 0331.05004
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