Covering systems, Kubert identities and difference equations

Štefan Porubský

Mathematica Slovaca (2000)

  • Volume: 50, Issue: 4, page 381-413
  • ISSN: 0139-9918

How to cite

top

Porubský, Štefan. "Covering systems, Kubert identities and difference equations." Mathematica Slovaca 50.4 (2000): 381-413. <http://eudml.org/doc/34521>.

@article{Porubský2000,
author = {Porubský, Štefan},
journal = {Mathematica Slovaca},
keywords = {covering system; Bernoulli polynomial; Hurwitz zeta function; Lerch function; difference equation; Kubert identities; Euler polynomial},
language = {eng},
number = {4},
pages = {381-413},
publisher = {Mathematical Institute of the Slovak Academy of Sciences},
title = {Covering systems, Kubert identities and difference equations},
url = {http://eudml.org/doc/34521},
volume = {50},
year = {2000},
}

TY - JOUR
AU - Porubský, Štefan
TI - Covering systems, Kubert identities and difference equations
JO - Mathematica Slovaca
PY - 2000
PB - Mathematical Institute of the Slovak Academy of Sciences
VL - 50
IS - 4
SP - 381
EP - 413
LA - eng
KW - covering system; Bernoulli polynomial; Hurwitz zeta function; Lerch function; difference equation; Kubert identities; Euler polynomial
UR - http://eudml.org/doc/34521
ER -

References

top
  1. APOSTOL T. M., On the Lerch zeta function, Pacific J. Math. 1 (1951), 161-167 (Addendum to: On the Lerch zeta function, ibid 2 (1952). 10). (1951) Zbl0043.07103MR0043843
  2. ERDELYI A.-MAGNUS W.-OBERHETTINGER F.-TRICOMI F. G., Higher Transcendental Functions. Vol. I, (Bateman Manuscript Project), McGraw-Hill Book Co., New York-London-Toronto, 1953. (1953) Zbl0051.30303
  3. BEEBEE J., Exact covering systems, cyclic sequences, and circuits in the d-Cube, Manuscript 1978, Department of Mathematical Sciences, University of Alaska Anchorage, Anchorage AK 99508. (1978) 
  4. BEEBEE J., Some trigonometric identities related to exact covers, Proc. Amer. Math. Soc. 112 (1991), 329-338. (1991) Zbl0725.11008MR1049133
  5. BEEBEE J., Bernoulli numbers and exact covering systems, Amer. Math. Monthly 99 (1992), 946-948. (1992) Zbl0776.11008MR1190561
  6. BEEBEE J., Exact covering systems and the Gauß-Legendre multiplication formula for the Gamma function, Proc. Amer. Math. Soc. 120 (1994), 1061-1065. (1994) Zbl0801.11005MR1180463
  7. BEEBEE J., Exact covers and the Kubert identities, Manuscript 1995, Department of Mathematical Sciences, University of Alaska Anchorage. Anchorage AK 99508. (1995) 
  8. CARLITZ L., The multiplication formulas for the Bernoulli and Euler polynomials, Math. Mag. 27 (1953), 59-64. (1953) Zbl0051.30704MR0058020
  9. DEEBA E. Y.-RODRIGUES D. M., Stirling's series and Bernoulli numbers, Amer. Math. Monthly 98 (1991), 423-426. (1991) Zbl0743.11012MR1104307
  10. Bernoulli Numbers, (Bibliography 1713-1990) (Dilcher, K., Skula, L., Slavutskij, I. Sh., eds.) Queen's Papers in Pure and Appl. Math. 87, Queen's Univ., Kingston, ON, 1991. (1990) Zbl0746.11013MR1119305
  11. EULER L., Introductio in analysin infinitorum I, Bousquet & Socios., Lausanne, 1748. Zbl0096.00303
  12. EULER L., Institutiones calculi differentialis, Petrograd, 1755. Zbl44.0010.03
  13. FRAENKEL A. S., A characterization of exactly covering congruences, Discrete Math. 4 (1973), 359-366. (1973) Zbl0257.10003MR0316359
  14. FRAENKEL A. S., Further characterizations and properties of exactly covering congruences, Discrete Math. 12 (1975), 93-100. (1975) Zbl0306.10002MR0374076
  15. FROBENIUS G., Über die Bernoulli'schen Zahlen und die Euler'schen Polynome, Sitzungsber. Preuss. Akad. Wiss. (1910), 809-847. (1910) Zbl1264.11013
  16. HURWITZ A., Einige Eigenschaften der Dirichlet’schen Functionen F ( s ) = Σ ( D / n ) · 1 / n s , die bei der Bestimmung der Classenanzahl binarer quadratischer Formen auftreten, Z. Math. Phys. 27 (1882), 86-101. 
  17. KOREC I., Irreducible disjoint covering systems, Acta Arit. 44 (1984), 389-395. (1984) Zbl0519.10002MR0777015
  18. LANG S., Cyclotomic Fields, Springer Verlag, New York-Heidelberg-Berlin, 1978. (1978) Zbl0395.12005MR0485768
  19. LERCH M., Note sur la fonction k ( w , x , s ) = k = 0 g 2 k π i x ( w + s ) x , Acta Math. 11 (1887-88), 19-24. MR1554747
  20. LERCH M., Foundation of the theory of Malmsten's series, C. R. Czech Acad. Sci. Prague 1 (1892), 1-68. (Czech) Zbl30.0256.01
  21. LERCH M., Investigations in the theory of Malmsten's series and invariants of quadratic forms, C R. Czech Acad. Sci. Prague 2 (1893), 1-11. (Czech) 
  22. LERCH M., Further investigation of Malmsten's series, C. R. Czech Acad. Sci. Prague 3 (1894), 1-61 (cf. Jahrbuch Über die Fortschritte der Math. 25 (1893-94), p. 484). (Czech) Zbl30.0256.01
  23. LINDELÖF E., Le calcul des résidues et ses applications a la theorie des fonctions, Gauthier-Villars, Paris, 1905. (1905) Zbl36.0468.01
  24. LIPSCHITZ R., Untersuchung einer aus vier Elementen gebildeten Reihe, J. Reine Angew. Math. 54 (1857), 313-328. 
  25. MALMSTEN, C J., De integralibus quisbusdam definitis, seriebusque infinitis, J. Reine Angew. Math. 38 (1849), 1-39. 
  26. MILNOR J., On polylogarithms, Hurwitz zeta functions, and the Kubert identities, Enseign. Math. (2) 29 (1983), 281-322. (1983) Zbl0557.10031MR0719313
  27. NIELSEN N., Traité Élémentaire des Nombres de Bernoulli, Gauthier-Villars, Paris, 1923. (1923) Zbl50.0170.04
  28. NOVÁK B.-ZNÁM Š., Disjoint covering systems, Amer. Math. Monthly 81 (1974), 42-45. (1974) Zbl0279.10043MR0332632
  29. NÖRLUND N. E., Vorlesungen über Differenzenrechnung, Springer Verlag, Berlin, 1924. (1924) Zbl50.0315.02
  30. NÖRLUND N. E., Sur la 'Somme' ďune Fonction, Gauthier-Villars, Paris, 1927. (1927) Zbl53.0430.08
  31. PORUBSKÝ Š., Natural exactly covering systems of congruences, Czechoslovak Math. J. 99 (1974), 598-606. (1974) Zbl0327.10005MR0354520
  32. PORUBSKY Š., Covering systems and generating functions, Acta Arith. 26 (1975), 223-231. (1975) Zbl0268.10044MR0379423
  33. PORUBSKÝ Š., On m times covering systems of congruences, Acta Arith. 29 (1976), 159-169. (1976) Zbl0282.10033MR0399033
  34. PORUBSKÝ Š., Results and problems on covering systems of residue classes, Mitt. Math. Sem. Giessen 150 (1981). (1981) Zbl0479.10032MR0638657
  35. PORUBSKÝ Š., A characterization of finite unions of arithmetic sequences, Discrete Math. 38 (1982), 73-77. (1982) Zbl0474.05001MR0676521
  36. PORUBSKÝ Š., Identities involving covering systems I, Math. Slovaca 44 (1994), 153-162. (1994) Zbl0809.11012MR1282532
  37. PORUBSKÝ Š., Identities involving covering systems II, Math. Slovaca 44 (1994), 555-568. (1994) Zbl0821.11008MR1338428
  38. PORUBSKÝ Š., Identities with covering systems and Appell polynomials, In: Number Theory in Progress. Procedings of the International conference on Number Theory organized by the Stefan Banach International Mathematical Center in Honor of the б0th Birthday of Andrzej Schinzel, Zakopane 1997, Walter de Gruyter, Berlin-New York, 1999, pp. 407-417. (1997) Zbl0934.11009MR1689520
  39. RAABE J. L., Die Jacob Bernoull'ische Function, Zürich, 1848, pp. 23-28. 
  40. STEIN S. K., Unions of arithmetic sequences, Math. Ann. 134 (1958), 289-294. (1958) Zbl0084.04302MR0093493
  41. SUN Z., Systems of congruences with multiplicators, (English summary), J. Nanjing Univ. Math. Biq. 6 (1989), 124-133. (Chinese) (1989) Zbl0703.11002MR1027346
  42. SUN Z., Several results on systems of residue classes, (Research announcement), Adv. in Math. (China) 18 (1989), 251-252. (1989) 
  43. WALUM H., Multiplication formulae for periodic functions, Pacific J. Math. 149 (1991), 383-396. (1991) Zbl0736.11012MR1105705
  44. ZNÁM Š., A simple characterization of disjoint covering systems, Discrete Math. 12 (1975), 89-91. (1975) Zbl0306.10001MR0376597

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.