A　numerical　approach　for　analyzing　the　guided　wave　propagation　behavior　in　a　functionally　graded　material　(FGM)　hollow　cylinder　is　presented.　Based　on　the　three-dimensional　linear　elasticity　theory,　the　state　matrix　form　of　displacement　and　stress　component　is　introduced　into　the　differential　governing　equation.　Combining　with　boundary　conditions,　the　dispersion　characteristic　equations　can　be　constructed.　Meanwhile,　taking　the　orthogonal　completeness　of　Legendre　series　into　consideration,　the　exhausting　root　findings　of　transcendental　equation　can　be　avoided,　and　it　transforms　the　redundant　integral　operators　into　analytical　expressions　by　means　of　its　recursive　properties.　Moreover,　it　is　not　only　unnecessary　to　stratify　FGMs,　and　introducing　the　univariate　nonlinear　regression　to　modelling　the　arbitrary　gradient　distribution,　and　transforming　the　redundant　integral　operator　into　an　analytical　expression　by　means　of　recursive　property.　It　should　be　pointed　out　that　the　coefficient　matrix　consists　only　zero　and　constant　terms,　which　are　related　to　the　elastic　constants　and　normalized　wave　number.　And　it　provides　a　fast　and　effectively　approach　to　extracting　the　dispersion　curves,　displacement　distribution　and　stress　profile,　simultaneously.　Meanwhile,　the　proposed　method　can　get　rid　of　the　exhausted　root-finding　algorithm,　and　accordingly　distinguish　the　different　modes　and　the　dispersion　curves　simultaneously.　Finally,　the　typical　non-stratified　computation　of　dispersion　curves　of　longitudinal　guided　wave　for　FGM　hollow　cylinder　is　realized.　The　accuracy　of　the　proposed　method　is　further　demonstrated　through　comparison　with　the　available　data　from　Legendre　polynomial　method.　Several　numerical　cases　about　iron　based　alumina　FGM　cylinder　are　studied,　and　the　convergence　of　the　present　polynomial　approach　is　discussed.　Then　the　effect　of　the　circumferential　wave　number,　gradient　variation,　diameter-thickness　ratio　on　the　dispersion　characteristics　are　illustrated.　Moreover,　the　distribution　characteristics　of　displacement　and　stress　profiles　of　F(1,1)　mode　at　a　given　frequency　are　discussed　in　detail.　It　is　shown　that　the　state-vector　formalism　and　Legendre　polynomial　hybrid　approach　is　a　powerful　tool　in　solving　wave　motions　in　FGM　elastic　cylinder,　which　lays　a　theoretical　foundation　for　the　non-destructive　evaluation　and　quantitative　characterization　of　the　gradient　structure　characteristics　of　functionally　graded　hollow　cylinder.