Some new transformations for Bailey pairs and WP-Bailey pairs

James Mc Laughlin

Open Mathematics (2010)

  • Volume: 8, Issue: 3, page 474-487
  • ISSN: 2391-5455

Abstract

top
We derive several new transformations relating WP-Bailey pairs. We also consider the corresponding transformations relating standard Bailey pairs, and as a consequence, derive some quite general expansions for products of theta functions which can also be expressed as certain types of Lambert series.

How to cite

top

James Mc Laughlin. "Some new transformations for Bailey pairs and WP-Bailey pairs." Open Mathematics 8.3 (2010): 474-487. <http://eudml.org/doc/268987>.

@article{JamesMcLaughlin2010,
abstract = {We derive several new transformations relating WP-Bailey pairs. We also consider the corresponding transformations relating standard Bailey pairs, and as a consequence, derive some quite general expansions for products of theta functions which can also be expressed as certain types of Lambert series.},
author = {James Mc Laughlin},
journal = {Open Mathematics},
keywords = {Bailey pairs; WP-Bailey Chains; WP-Bailey pairs; Lambert Series; Basic Hypergeometric Series; q-series; Theta series; WP-Bailey chains; Lambert series; basic hypergeometric series; theta series},
language = {eng},
number = {3},
pages = {474-487},
title = {Some new transformations for Bailey pairs and WP-Bailey pairs},
url = {http://eudml.org/doc/268987},
volume = {8},
year = {2010},
}

TY - JOUR
AU - James Mc Laughlin
TI - Some new transformations for Bailey pairs and WP-Bailey pairs
JO - Open Mathematics
PY - 2010
VL - 8
IS - 3
SP - 474
EP - 487
AB - We derive several new transformations relating WP-Bailey pairs. We also consider the corresponding transformations relating standard Bailey pairs, and as a consequence, derive some quite general expansions for products of theta functions which can also be expressed as certain types of Lambert series.
LA - eng
KW - Bailey pairs; WP-Bailey Chains; WP-Bailey pairs; Lambert Series; Basic Hypergeometric Series; q-series; Theta series; WP-Bailey chains; Lambert series; basic hypergeometric series; theta series
UR - http://eudml.org/doc/268987
ER -

References

top
  1. [1] Andrews G.E., Bailey’s transform, lemma, chains and tree, Special functions 2000: current perspective and future directions (Tempe, AZ), 1–22, NATO Sci. Ser. II Math. Phys. Chem., 30, Kluwer Acad. Publ., Dordrecht, 2001 Zbl1005.33005
  2. [2] Andrews G.E., Berkovich A, The WP-Bailey tree and its implications, J. London Math. Soc.(2), 2002, 66(3), 529–549 http://dx.doi.org/10.1112/S0024610702003617 Zbl1049.33010
  3. [3] Andrews G.E., Berndt B.C., Ramanujans Lost Notebook, Part I, Springer, 2005 Zbl1075.11001
  4. [4] Andrews G.E., Lewis R., Liu Z.G., An identity relating a theta function to a sum of Lambert series, Bull. London Math. Soc., 2001, 33(1), 25–31 http://dx.doi.org/10.1112/blms/33.1.25 Zbl1021.11010
  5. [5] Berndt B.C., Ramanujans Notebooks, Part III, Springer-Verlag, New York, 1991 Zbl0733.11001
  6. [6] Berndt B.C., Ramanujans Notebooks, Part V, Springer-Verlag, New York, 1998 
  7. [7] Borwein J.M., Borwein P.B., A cubic counterpart of Jacobi’s identity and the AGM, Trans. Amer. Math. Soc, 1991, 323(2), 691–701 http://dx.doi.org/10.2307/2001551 Zbl0725.33014
  8. [8] Bressoud D., Some identities for terminating q-series, Math. Proc. Cambridge Philos. Soc, 1981, 89(2), 211–223 http://dx.doi.org/10.1017/S0305004100058114 Zbl0454.33003
  9. [9] Gasper G., Rahman M., Basic hypergeometric series, With a foreword by Richard Askey, Encyclopedia of Mathematics and its Applications, 96, Cambridge University Press, Cambridge, 2004 Zbl1129.33005
  10. [10] Liu Q., Ma X., On the Characteristic Equation of Well-Poised Baily Chains, Ramanujan J., 2009, 18(3), 351–370 http://dx.doi.org/10.1007/s11139-007-9060-6 Zbl1172.05009
  11. [11] Mc Laughlin J., Sills A.V., Zimmer P., Some implications of Chu’s 10Ψ10 extension of Bailey’s 6Ψ6 summation formula, preprint 
  12. [12] Mc Laughlin J., Zimmer P., General WP-Bailey Chains, Ramanujan J., 2010, 22(1), 11–31 http://dx.doi.org/10.1007/s11139-010-9220-y 
  13. [13] Mc Laughlin J., Zimmer P., Some Implications of the WP-Bailey Tree, Adv. in Appl. Math., 2009, 43(2), 162–175 http://dx.doi.org/10.1016/j.aam.2009.02.001 Zbl1173.33015
  14. [14] Singh U.B., A note on a transformation of Bailey, Quart. J. Math. Oxford Ser. (2), 1994, 45(177), 111–116 http://dx.doi.org/10.1093/qmath/45.1.111 
  15. [15] Slater L.J., A new proof of Rogers’s transformations of infinite series, Proc. London Math. Soc. (2), 1951, 53, 460–475 http://dx.doi.org/10.1112/plms/s2-53.6.460 Zbl0044.06102
  16. [16] Spiridonov V.P., An elliptic incarnation of the Bailey chain, Int. Math. Res. Not., 2002, 37, 1945–1977 http://dx.doi.org/10.1155/S1073792802205127 Zbl1185.33023
  17. [17] Watson G.N., The Final Problem: An Account of the Mock Theta Functions, J. London Math. Soc., 1936, 11, 55–80 http://dx.doi.org/10.1112/jlms/s1-11.1.55 Zbl0013.11502
  18. [18] Warnaar S.O, Extensions of the well-poised and elliptic well-poised Bailey lemma, Indag. Math. (N.S.), 2003, 14, 571–588 http://dx.doi.org/10.1016/S0019-3577(03)90061-9 Zbl1054.33013

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.