@inproceedings {MeWi2016, 
AUTHOR = {Stephen Melczer and Wilson, Mark C.},
     TITLE = {Asymptotics of lattice walks via analytic combinatorics in several variables},
 BOOKTITLE = {2016 {C}onference on {F}ormal {P}ower {S}eries and {A}lgebraic {C}ombinatorics, FPSAC2016},
    SERIES = {Discrete Math. Theor. Comput. Sci. Proc., AH},
     PAGES = {863-874},
 PUBLISHER = {Assoc. Discrete Math. Theor. Comput. Sci., Nancy},
      YEAR = {2016},
  keywords={ACSV applications},
  url_Paper={https://dmtcs.episciences.org/6390/pdf},
  abstract={We consider the enumeration of walks on the two dimensional non-negative
integer lattice with short steps. Up to isomorphism there are 79 unique
two dimensional models to consider, and previous work in this area has
used the kernel method, along with a rigorous computer algebra approach,
to show that 23 of the 79 models admit D-finite generating functions. In
2009, Bostan and Kauers used Pad\'{e}-Hermite approximants to guess
differential equations which these 23 generating functions satisfy, in
the process guessing asymptotics of their coefficient sequences. In this
article we provide, for the first time, a complete rigorous verification
of these guesses. Our technique is to use the kernel method to express
19 of the 23 generating functions as diagonals of tri-variate rational
functions and apply the methods of analytic combinatorics in several
variables (the remaining 4 models have algebraic generating functions
and can thus be handled by univariate techniques). This approach also
shows the link between combinatorial properties of the models and
features of its asymptotics such as asymptotic and polynomial growth
factors. In addition, we give expressions for the number of walks
returning to the x-axis, the y-axis, and the origin, proving recently
conjectured asymptotics of Bostan, Chyzak, van Hoeij, Kauers, and Pech.}
  
}

{"_id":"qHfJQZCtQaiBBZDDE","bibbaseid":"melczer-wilson-asymptoticsoflatticewalksviaanalyticcombinatoricsinseveralvariables-2016","author_short":["Melczer, S.","Wilson, M. C."],"bibdata":{"bibtype":"inproceedings","type":"inproceedings","author":[{"firstnames":["Stephen"],"propositions":[],"lastnames":["Melczer"],"suffixes":[]},{"propositions":[],"lastnames":["Wilson"],"firstnames":["Mark","C."],"suffixes":[]}],"title":"Asymptotics of lattice walks via analytic combinatorics in several variables","booktitle":"2016 Conference on Formal Power Series and Algebraic Combinatorics, FPSAC2016","series":"Discrete Math. Theor. Comput. Sci. Proc., AH","pages":"863-874","publisher":"Assoc. Discrete Math. Theor. Comput. Sci., Nancy","year":"2016","keywords":"ACSV applications","url_paper":"https://dmtcs.episciences.org/6390/pdf","abstract":"We consider the enumeration of walks on the two dimensional non-negative integer lattice with short steps. Up to isomorphism there are 79 unique two dimensional models to consider, and previous work in this area has used the kernel method, along with a rigorous computer algebra approach, to show that 23 of the 79 models admit D-finite generating functions. In 2009, Bostan and Kauers used Padé-Hermite approximants to guess differential equations which these 23 generating functions satisfy, in the process guessing asymptotics of their coefficient sequences. In this article we provide, for the first time, a complete rigorous verification of these guesses. Our technique is to use the kernel method to express 19 of the 23 generating functions as diagonals of tri-variate rational functions and apply the methods of analytic combinatorics in several variables (the remaining 4 models have algebraic generating functions and can thus be handled by univariate techniques). This approach also shows the link between combinatorial properties of the models and features of its asymptotics such as asymptotic and polynomial growth factors. In addition, we give expressions for the number of walks returning to the x-axis, the y-axis, and the origin, proving recently conjectured asymptotics of Bostan, Chyzak, van Hoeij, Kauers, and Pech.","bibtex":"@inproceedings {MeWi2016, \nAUTHOR = {Stephen Melczer and Wilson, Mark C.},\n TITLE = {Asymptotics of lattice walks via analytic combinatorics in several variables},\n BOOKTITLE = {2016 {C}onference on {F}ormal {P}ower {S}eries and {A}lgebraic {C}ombinatorics, FPSAC2016},\n SERIES = {Discrete Math. Theor. Comput. Sci. Proc., AH},\n PAGES = {863-874},\n PUBLISHER = {Assoc. Discrete Math. Theor. Comput. Sci., Nancy},\n YEAR = {2016},\n keywords={ACSV applications},\n url_Paper={https://dmtcs.episciences.org/6390/pdf},\n abstract={We consider the enumeration of walks on the two dimensional non-negative\ninteger lattice with short steps. Up to isomorphism there are 79 unique\ntwo dimensional models to consider, and previous work in this area has\nused the kernel method, along with a rigorous computer algebra approach,\nto show that 23 of the 79 models admit D-finite generating functions. In\n2009, Bostan and Kauers used Pad\\'{e}-Hermite approximants to guess\ndifferential equations which these 23 generating functions satisfy, in\nthe process guessing asymptotics of their coefficient sequences. In this\narticle we provide, for the first time, a complete rigorous verification\nof these guesses. Our technique is to use the kernel method to express\n19 of the 23 generating functions as diagonals of tri-variate rational\nfunctions and apply the methods of analytic combinatorics in several\nvariables (the remaining 4 models have algebraic generating functions\nand can thus be handled by univariate techniques). This approach also\nshows the link between combinatorial properties of the models and\nfeatures of its asymptotics such as asymptotic and polynomial growth\nfactors. In addition, we give expressions for the number of walks\nreturning to the x-axis, the y-axis, and the origin, proving recently\nconjectured asymptotics of Bostan, Chyzak, van Hoeij, Kauers, and Pech.}\n \n}\n\n\n","author_short":["Melczer, S.","Wilson, M. 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