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one of the motivations of this paper is the asymptotic analysis of the generalized laguerre polynomials , denoted by @xmath7 , with complex varying parameters , whose definition and properties can be found for instance in chapter v of szeg s classic memoir @xcite . they can be given explicitly by @xmath8 or , equivalently , by the well - known rodrigues formula @xmath9\,.\ ] ] expressions and make sense for complex values of the parameter @xmath1 , showing that @xmath7 depend analytically on @xmath1 . when indeterminacy occurs in evaluating the coefficients in , we understand them in the sense of their analytic continuation with respect to @xmath1 . with this convention we see that @xmath10 so that @xmath11 for all @xmath12 . moreover , for any @xmath12 , @xmath7 is the unique ( up to a multiplicative constant ) polynomial solution of the differential equation @xmath13 which shows that every zero of @xmath7 different from @xmath14 must be simple . in fact , multiple zeros ( at the origin ) can appear if and only if @xmath15 . in this case the reduction formula @xmath16 shows that @xmath14 is a zero of @xmath17 of multiplicity @xmath18 , see again @xcite for details . orthogonality conditions satisfied by the laguerre polynomials can be easily derived from iterating integration by parts , see e.g.  @xcite ; we reproduce the arguments in section [ sec : orthogonality ] for the sake of completeness . the weight of orthogonality is the algebraic function @xmath19 and the integration goes along a contour in the complex plane . the classical situation is @xmath20 , in which case the orthogonality of @xmath21 reduces to @xmath22 as a consequence , for @xmath23 zeros of @xmath21 are positive and simple .    in this paper we study sequences of laguerre polynomials with , in general , complex parameters @xmath1 depending on the degree @xmath2 . more precisely , we assume that @xmath24 with these hypotheses , but only for _ real _ parameters @xmath25 , sequences @xmath26 were studied , in particular , in @xcite ( weak asymptotics ) and in @xcite ( strong asymptotics ) . this paper is a natural continuation of this study , although complex values of @xmath6 introduce an asymmetry that makes the problem more difficult . the main ingredient of the asymptotic analysis is the right choice of the contour of orthogonality , which is related to the trajectories of an associated quadratic differential . precisely the description of the structure of these trajectories ( section [ sec : trajectories ] ) constitutes the core of our contribution . it is known that under assumptions we need to perform a linear scaling in the variable in order to fix the geometry of the problem . thus , we will study the sequence @xmath27 the zeros of @xmath28 cluster along certain curves in the complex plane , corresponding to trajectories ( known also as stokes lines ) of a quadratic differential , depending on a parameter @xmath6 , see figure  [ fig : cerostray ] . this is not surprising : as it follows from the pioneering works of stahl @xcite , and later of gonchar and rakhmanov @xcite , the support of the limiting zero - counting measure of such polynomials is a set of analytic curves exhibiting the so - called @xmath29-property . they can also be characterized as trajectories of a certain quadratic differential on the riemann surface . the explicit expression of the quadratic differential associated to polynomials @xmath28 can be easily derived from the differential equation , see @xcite for details .     for @xmath30 and @xmath31 . ]    it turns out that for @xmath32 , the support of the limiting zero - counting measure for @xmath28 s is a simple analytic and real - symmetric arc , which as @xmath6 approaches @xmath33 , closes itself to form for @xmath34 the well - known szeg curve @xcite . when @xmath35 , the support becomes an interval of the positive semi - axis . case @xmath36 is special : generically , the support is connected , and consists of a closed loop surrounding the origin together with an interval of the positive semi - axis . however , when @xmath25 s are exponentially close to integers , the support can split into two disjoint components , the closed contour and the interval , see @xcite for details .    in the case @xmath37 the trajectories of these quadratic differentials , and subsequently , the support of the limiting zero - counting measure for @xmath28 s , have not been described ; this is done in section [ sec : trajectories ] , which as we pointed out , is probably one of the central contributions of this paper .    being the generalized laguerre polynomials such a classical object , it is not surprising that their asymptotics has been well studied , using different and complementary approaches based on many characterizations of these polynomials . many of these results correspond to the case @xmath38 , when all their zeros belong to the positive semi - axis , see e.g.  @xcite . others study the asymptotics with a _ fixed _ parameter @xmath1 , such as in @xcite , where additionally many interdisciplinary applications of laguerre asymptotics are explained . the gonchar - rakhmanov theory was used to find the weak asymptotics ( or asymptotic zero distribution ) of @xmath28 s for general @xmath5 satisfying , see @xcite ; we extend it to complex @xmath25 s in this work , see section  [ sec : results ] . the critical case @xmath39 in this context was analyzed in @xcite , using the extremality of the family of polynomials . the non - linear steepest descent method of deift  zhou introduced in @xcite , and further developed in @xcite and @xcite ( see also @xcite ) , based on the riemann hilbert characterization of orthogonality by fokas , its , and kitaev @xcite , is an extremely powerful technique , rendering exhaustive answers in cases previously intractable . following this approach , kuijlaars and mclaughlin @xcite found the strong asymptotics in the whole complex plane for the family @xmath40 and arbitrary values of @xmath41 . the crucial ingredient of this asymptotic analysis is the choice of an appropriate path of integration on the complex plane , based on the structure of the trajectories of the associated quadratic differential . taking advantage of the results of section [ sec : trajectories ] we prove the existence and describe such paths of integration , which allows us to carry out the steepest descent analysis in the spirit of kuijlaars and mclaughlin . as a result , we obtain in section  [ sec : results ] the detailed strong asymptotics for the rescaled laguerre polynomials @xmath28 . this also sheds light on one of the open questions mentioned in @xcite . the generalized bessel polynomials @xmath42 can be defined as @xmath43 the asymptotic distribution of their zeros and their strong asymptotics is a straightforward consequence of theorems [ teolag1 ] and [ thm : stronga ] below , by replacing @xmath44 and @xmath45 . let @xmath46 be a complex parameter , for which we define the monic polynomials @xmath47 with @xmath48 since @xmath49 is real - symmetric with respect to the parameter @xmath6 , without loss of generality we can assume in what follows that @xmath50 , and that the square root in stands for its main branch in the closed upper half plane . in what follows , we use the notation @xmath51 while as usual , @xmath52 and @xmath53 stand for the open positive and negative real semi - axes , respectively .    on the riemann sphere @xmath54 we define the quadratic differential @xmath55 written in the natural parametrization of the complex plane . its _ horizontal trajectories _ ( or just trajectories in the future ) are the loci of the equation @xmath56 the _ vertical _ or _ orthogonal _ trajectories are obtained by replacing @xmath57 by @xmath58 in the equation above . the trajectories and the orthogonal trajectories of @xmath59 produce a transversal foliation of the riemann sphere @xmath54 .    in order to study the global structure of these trajectories on the plane we start by observing that @xmath59 has two zeros , @xmath60 , that are distinct and simple if and only if @xmath61 , and a double pole at the origin if @xmath62 , with @xmath63 another pole of @xmath59 is located at infinity and is of order 4 ; with the parametrization @xmath64 , @xmath65 points from @xmath66 are regular . the local structure of the trajectories is well known ( see e.g.  @xcite , @xcite , @xcite , @xcite ) . at any regular point trajectories look locally as simple analytic arcs passing through this point , and through every regular point of @xmath59 passes a uniquely determined horizontal and uniquely determined vertical trajectory of @xmath59 , that are locally orthogonal at this point ( * ? ? ? * theorem 5.5 ) .    for @xmath67 , there are @xmath68 trajectories emanating from @xmath60 under equal angles @xmath69 . in the case of the origin , the trajectories have either the radial , the circular or the log - spiral form , depending on the vanishing of the real or imaginary part of @xmath6 , see figure [ fig : localstructure2 ] .    [ cols= " < , < , < " , ]     the boundary of @xmath70 consists of critical trajectories only , so that using the definition of @xmath71 it is straightforward to see that @xmath72 is the left half plane . moreover , by , @xmath73 and the @xmath74 side of the short trajectory @xmath75 is mapped onto the vertical segment @xmath76 $ ] .    on the other hand , the `` @xmath77 '' side of @xmath78 and the `` @xmath74 '' side of @xmath79 ( oriented from the origin to @xmath80 ) are mapped onto the imaginary axis . there are two asymptotic directions to @xmath80 from @xmath81 ; for one of them ( from the `` @xmath74 '' side of @xmath79 ) by definition @xmath82 for the other direction , from the `` @xmath77 '' side of @xmath79 , @xmath83 where the contour of integration encircles the origin in the anti - clockwise direction , and where we have used . observe that @xmath84 for @xmath85 . thus , the boundary @xmath86 is mapped onto the vertical line in the right half plane , passing through @xmath87 . by , @xmath88 so that @xmath89 corresponds by @xmath90 to the vertical segment @xmath91 $ ] . this concludes the proof of . the corresponding results for function @xmath92 are established using the following connection formula : @xmath93          \widetilde{\phi}(z ) + \pi i(1+a ) , & \quad \text{for } z \in \omega_- ,      \end{array } \right.\ ] ] which is a direct consequence of formulas , and the definition of @xmath92 : @xmath94    observe also that with proposition [ prop : confmapping ] we conclude the proof of theorem [ thm:1 ] . our main result of the first part of this section is the following    [ prop : progressive ] there exist two analytic curves , @xmath95 and @xmath96 , in @xmath97 , emanating from @xmath98 and @xmath80 , respectively , and diverging to infinity in the asymptotic direction @xmath99 , and such that @xmath100    from the considerations at the end of section  [ sec : trajectories ] it follows that one instance of @xmath95 could be the orthogonal trajectory of @xmath59 emanating from @xmath98 and diverging in the @xmath99 direction , see figure  [ fig : globalstructure ] or figure  [ fig : trajectorieswithzeros ] below . however , this fact is not so relevant ; what really matters for our further analysis is the existence of both curves stated in this corollary .    in what follows we will make use of the contour @xmath101 oriented clockwise , in such a way that @xmath102 is not homotopic to a point in @xmath103 , and the origin remains on the right of the curve . observe that @xmath102 is not uniquely determined due to the freedom in the choice of @xmath104 , although @xmath75 is . we turn now to the second goal of this section . the main tools for the study of the weak asymptotic behavior of polynomials satisfying a non - hermitian orthogonality were developed in the seminal works of stahl @xcite and gonchar and rakhmanov @xcite . they showed that when the complex analytic weight function depends on the degree of the polynomial , the limit zero distribution is characterized by an equilibrium problem on a compact set in the presence of an external field ; this compact set must satisfy a symmetry property with respect to the external field .    for any positive borel measure @xmath105 on @xmath106 , such that @xmath107 given @xmath6 , @xmath108 , consider the harmonic _ external field _ @xmath109 in @xmath103 , where we take the main branches of @xmath110 and @xmath111 . let also the contour @xmath102 be as defined in . it is known that the ( unique ) probability _ equilibrium measure _ @xmath105 on @xmath102 in the external field @xmath112 can be characterized by the property @xmath113 where @xmath114 is the equilibrium constant ; for details see e.g.  @xcite or @xcite . both the measure @xmath105 and its support @xmath115 have the _ @xmath29-property _ in the external field @xmath112 if at any interior point @xmath116 of @xmath115 , @xmath117 where @xmath118 are the normals to @xmath115 . [ prop : eq_measure ] the absolutely continuous measure @xmath119 is the equilibrium measure on @xmath102 in the external field @xmath112 . function @xmath120 was defined in . recall that as usual , @xmath75 is oriented from @xmath80 to @xmath98 , and @xmath121 is the boundary value on its left side .    by the definition of @xmath75 , the right hand side in is real - valued and non - vanishing along @xmath75 . taking into account we conclude that @xmath105 is a probability measure on @xmath75 . furthermore , as in the proof of lemma  [ lemma : integrals ] and using , for @xmath122 , @xmath123 thus , @xmath124 thus , for @xmath125 , @xmath126 where @xmath92 was defined in , and @xmath127 is an appropriately chosen real constant . it remains to use that the last term in the right hand side vanishes on @xmath75 and is strictly positive on @xmath104 ( see corollary  [ prop : progressive ] ) to conclude that for @xmath105 in characterization holds . finally , since @xmath128 , harmonic in @xmath129 , is identically @xmath130 on @xmath131 and is the real part of the analytic function @xmath132 satisfying @xmath133 on @xmath75 , the symmetry property easily follows from the cauchy  riemann equations .    in the next section a relevant role will be played by the so - called @xmath134-function , i.e.  the complex potential of the equilibrium measure @xmath135 on @xmath75 : @xmath136 where for each @xmath137 we view @xmath138 as an analytic function of the variable @xmath139 , with branch cut emanating from @xmath140 ; the cut is taken along @xmath141 . now we have all ingredients to formulate and prove the asymptotic results for the rescaled generalized laguerre polynomials @xmath28 , defined in in section [ sec : intro ] , under the assumption . the analysis is based on the non - hermitian orthogonality conditions satisfied by these polynomials ( section [ sec : orthogonality ] ) . for the weak asymptotics ( or limiting zero distribution ) we can use an analogue of the gonchar stahl s results , while the strong asymptotics is derived from the corresponding riemann - hilbert characterization of these polynomials ( section [ sec : orthogonality ] ) using the non - linear steepest descent method of deift and zhou .      throughout this section we assume that @xmath142 . then , the generalized laguerre polynomials @xmath7 satisfy a non - hermitian orthogonality in the complex plane , see @xcite . namely ,    [ thort ] let @xmath143 in @xmath144 be an unbounded jordan arc diverging in both directions toward @xmath145 , @xmath146 , and @xmath142 . then @xmath147 in addition , @xmath148    although the selection of the branch of @xmath149 is not relevant , for the sake of definiteness we take here @xmath150 , with @xmath151 .    in the proof we use @xmath152 to denote the @xmath18-th derivative of @xmath153 and @xmath154 .    by the rodrigues formula ( [ rodrlag ] ) , @xmath155 integrating in the left hand side of @xmath2 times by parts and using ( [ rodrlag2 ] ) , we get @xmath156^{(j ) }     w^{(n - j-1)}(z ; n+\alpha )   \bigg|_{\sigma } \\   + \frac{1}{n ! } \ , \int_{\sigma } \big[z^k \big]^{(n ) } w(z ; n+\alpha)\ ,   dz\ , . \end{split}\ ] ] since @xmath157 is single - valued on @xmath143 , @xmath158^{(j ) }     w^{(n - j-1)}(z ; n+\alpha ) \bigg|_{\sigma}=0\ , , \qquad \mbox{for } 0 \leq j \leq n-1\,.\ ] ] thus , if @xmath159 , all the terms in the right - hand side of ( [ partes ] ) vanish , and ( [ eq21 ] ) follows .    furthermore , for @xmath160 , we also get @xmath161 we deform @xmath143 to the positive real axis to obtain @xmath162 where @xmath163 denotes the gamma function . by analytic continuation the integral in ( [ eq22 ] ) is equal to ( [ eq23 ] ) for every @xmath1 , and ( [ eq22 ] ) follows . .,width=944 ]    consider now the monic rescaled polynomials @xmath164 by theorem [ thort ] , @xmath165 verifies the orthogonality @xmath166      \neq 0 , & \quad \mbox { for } k = n ,      \end{array } \right.\ ] ] for a contour @xmath143 specified in theorem [ thort ] ( see figure  [ fig : curvaortog ] ) . by the classical work of fokas , its , and kitaev @xcite , this yields a characterization of the polynomial @xmath167 in terms of a riemann hilbert problem : determine a @xmath168 matrix valued function @xmath169 satisfying the following conditions :    1 . @xmath170 is analytic for @xmath171 , 2 . @xmath170 possesses continuous boundary values for @xmath172 . if @xmath143 is oriented clockwise , and @xmath173 , @xmath174 denote the non - tangential boundary values of @xmath170 on the left and right sides of @xmath143 , respectively , then @xmath175 3 . @xmath170 has the following behavior as @xmath176 : @xmath177    the unique solution of this problem for @xmath178 is given by ( see @xcite ) @xmath179      {   d_{n-1 } } p_{n-1}(z ) & \displaystyle{\frac{d_{n-1}}{2\pi i }   \int_{\sigma } }       \frac{p_{n-1}(\zeta ) \zeta^{\alpha } e^{-n\zeta}}{\zeta -z } \ , d\zeta       \end{pmatrix},\ ] ] where @xmath180 is the monic generalized laguerre polynomial ( [ eq24 ] ) and the constant @xmath181 is chosen to guarantee that for the @xmath182 entry of @xmath178 , @xmath183 the orientation of @xmath143 is consistent also with the one used in section [ sec : trajectories ] . with each @xmath28 in we can associate its normalized zero - counting measure @xmath184 , such that for any compact set @xmath185 in @xmath106 , @xmath186 ( the zeros are considered taking account of their multiplicity ) . weak asymptotics for @xmath28 s studies convergence of the sequence @xmath187 in the weak-@xmath188 topology , that we denote by @xmath189 . [ teolag1 ] let the sequence of ( generalized ) laguerre polynomials @xmath28 in ( [ sequencex ] ) satisfy ( [ limits ] ) with @xmath190 . then @xmath191 where @xmath105 is the probability measure given in proposition  [ prop : eq_measure ] . its support is a simple analytic arc , namely the short trajectory @xmath75 described in theorem  [ thm:1 ] .    regarding the strong asymptotics we have a result very similar to that in @xcite , but referring now to the short trajectory @xmath75 . because of the @xmath2-dependence of @xmath25 , all of the notions and results introduced previously are @xmath2-dependent . for example , we have that the curves @xmath192 are all varying with @xmath2 , and so we denote them by @xmath193 ; clearly , they tend to the limiting curve @xmath75 . likewise , we have that the functions @xmath194 , @xmath134 , @xmath92 , and @xmath195 , as well as all matrix - valued functions are @xmath2-dependent , and we also use a subscript @xmath2 to denote their dependence on @xmath2 , using the notation without subscript @xmath2 when referring to the limiting case .    [ thm : stronga ] for the rescaled generalized laguerre polynomials @xmath28 as @xmath196 ,    1 .   * ( asymptotics away from @xmath75 . ) * + uniformly for @xmath139 in compact subsets of @xmath197 , we have as @xmath196 , @xmath198 2 .   * ( asymptotics on @xmath74-side of @xmath193 , away from endpoints . ) * + uniformly for @xmath139 on the @xmath74-side of @xmath193 away from @xmath60 , we have as @xmath196 , @xmath199 .     \ ] ] 3 .   * ( asymptotics on @xmath77-side of @xmath193 , away from endpoints . ) * + uniformly for @xmath139 on the @xmath77-side of @xmath193 away from @xmath60 , we have as @xmath196 , @xmath200 .   \ ] ] 4 .   * ( asymptotics near @xmath98 . ) * + uniformly for @xmath139 in a ( small ) neighborhood of @xmath98 , we have as @xmath196 , @xmath201,\end{aligned}\ ] ] where @xmath202^{2/3}.\ ] ]    recall that @xmath203 was introduced in , while @xmath134 was defined in . see section  [ sec : proofs ] below for a more detailed explanation about the conformal mapping @xmath153 . we can write a similar asymptotic formula at the other endpoint of @xmath75 ; it will be in terms of the function @xmath71 appropriately redefined in a neighborhood @xmath204 of @xmath80 in such a way that it is analytic in @xmath205 . namely , by , we should use there @xmath206 where again @xmath79 is oriented from the origin to @xmath80 .    as a consequence of theorem  [ thm : stronga ] , we can make a stronger statement about the zero asymptotics of @xmath28 ( compare it with theorem  [ teolag1 ] ) :    [ corollary : zeros ] under assumptions of theorem  [ thm : stronga ] , for every neighborhood @xmath204 of @xmath75 , there is @xmath207 such that for every @xmath208 , all zeros of @xmath28 are in @xmath204 , and no zeros at the @xmath209 side of @xmath75 . figure  [ fig : trajectorieswithzeros ] is a good illustration of the statement of this corollary . zeros_and_trajectories ( 36,37)@xmath210 ( 28,28)@xmath211 ( 43,58)@xmath212 ( 64,71)@xmath213 ( 75,25)@xmath214 ( 24,46)@xmath75      all proofs are based on the complex non - hermitian varying orthogonality satisfied by the generalized laguerre polynomials with complex coefficients ( theorem [ thort ] ) . the cornerstone is the existence of the equilibrium measure @xmath105 having the @xmath29-property , established in proposition  [ prop : eq_measure ] . the connection of the weak asymptotics of non - hermitian orthogonal polynomials with the equilibrium with symmetry was established first by stahl @xcite ( for a fixed weight ) and extended by gonchar and rakhmanov to a varying orthogonality in @xcite . theorem  [ teolag1 ] is a straightforward consequence of the main theorem in @xcite , see also @xcite . on the other hand , theorem  [ thm : stronga ] is established using the non - linear steepest descent analysis of deif and zhou @xcite of the riemann hilbert ( rh ) problem described in section  [ sec : orthogonality ] , where we set @xmath215 , as defined in . the proof follows the scheme of the work  @xcite almost literally . thus , instead of repeating all the calculations step by step , we describe here the main transformations , referring the interested reader to  @xcite for details . it is convenient to redefine slightly function @xmath92 in , moving the branch cuts from @xmath78 to @xmath96 , and from @xmath79 to @xmath53 as follows . first , we continue @xmath92 analytically from @xmath70 through @xmath78 to the domain bounded by @xmath78 and @xmath96 . moreover , let @xmath216 be the arc of @xmath79 from @xmath80 to its first intersection with @xmath53 , see figure  [ fig : critical_gr_mod ] . then @xmath217 consists of two connected components : `` upper '' ( containing the origin ) and `` lower '' domains . hence , if we define @xmath218 it will be holomorphic in @xmath219 , see our analysis in section  [ sec : auxiliary ] .    as usual , the first transformation of the riemann - hilbert problem for @xmath178 is the regularization at infinity by means of the function @xmath134 introduced in  . by , there is a constant @xmath127 such that @xmath220 where @xmath221 is defined with a branch cut along @xmath53 . we define for @xmath222 , @xmath223 here , and in what follows , @xmath224 denotes the pauli matrix @xmath225 , so that for example @xmath226 .    from the riemann hilbert problem for @xmath178 it follows by a straightforward calculation that @xmath227 is the unique solution of the following rh problem : determine @xmath228 such that    1 . @xmath229 is analytic for @xmath222 , 2 . @xmath229 possesses continuous boundary values for @xmath230 , denoted by @xmath231 and @xmath232 , and @xmath233 for @xmath230 , 3 . @xmath229 behaves like the identity at infinity : @xmath234    the jump relation for @xmath227 has a different form on the three parts @xmath102 . using it is easy to obtain the following jump relations for @xmath134 across the contour @xmath102 ( see  @xcite ) : @xmath235 @xmath236 and @xmath237    the second transformation of the rh problem deals with the oscillatory behavior in the jump matrix for @xmath227 on @xmath75 , see , and is based on its standard factorization : @xmath238    as part of the steepest descent method we introduce the oriented contour @xmath239 , which consists of @xmath102 plus two simple curves @xmath240 from @xmath80 to @xmath98 , contained in @xmath70 and @xmath241 , respectively , as shown in figure  [ fig : lens1 ] . we choose @xmath240 such that @xmath242 on @xmath240 , which , as it follows from , is always possible . then @xmath243 has four connected components , denoted by @xmath244 , @xmath245 , @xmath246 , and @xmath247 as indicated in figure  [ fig : lens1 ] . lens1 ( 20,19)@xmath210 ( 10,32)@xmath248 ( -4,35)@xmath249 ( 20.5,35)@xmath250 ( 60,55)@xmath251 ( 37,52)@xmath212 ( 8,12)@xmath211 ( 51,7)@xmath252 ( 90,55)@xmath253 ( 9,42)@xmath254 ( 19,42)@xmath255 ( 50,40)@xmath256    consequently , we define @xmath257 by @xmath258 @xmath259 @xmath260 then from the rh problem for @xmath227 and the factorization ( [ eq47 ] ) we obtain that @xmath261 is the unique solution of the following riemann hilbert problem : determine a @xmath262 matrix valued function @xmath263 such that the following hold :    1 . @xmath264 is analytic for @xmath265 , 2 . @xmath264 possesses continuous boundary values for @xmath266 , denoted by @xmath267 and @xmath268 , and @xmath269 @xmath270 @xmath271 and @xmath272 3 . @xmath264 behaves like the identity at infinity : @xmath273    the global ( or outer ) parametrix corresponding to this problem is a matrix @xmath274 given by ( see e.g.  @xcite , ( * ? ? ? * section 7.3 ) , or ( * ? ? ? * section 5.1 ) ) @xmath275 where we take the main branches of the square roots . it is easy to verify that    1 . @xmath276 is analytic for @xmath277 , 2 . @xmath276 possesses continuous boundary values for @xmath278 , denoted by @xmath279 and @xmath280 , and @xmath281 3 . @xmath282 for @xmath176 . 4 .   near the endpoints @xmath60 it satisfies @xmath283    since the behavior of @xmath284 does not match the desired behavior at the endpoints @xmath60 of @xmath75 , we need one more construction around these points , namely the so - called local parametrix , well described for instance in @xcite .    from its definition in ( [ def : phi ] ) it is easy to see that the @xmath92-function has a convergent expansion @xmath285 in a neighborhood of @xmath98 . the factor @xmath286 is defined with a cut along @xmath141 . then @xmath153 , defined by @xmath287^{2/3},\ ] ] is analytic in a neighborhood of @xmath98 . we choose the @xmath288-root with a cut along @xmath75 and such that @xmath289 for @xmath290 . recall that @xmath291 on @xmath95 . then @xmath292 and @xmath293 . therefore we can deform @xmath95 locally and choose @xmath294 so small that @xmath295 is a one - to - one mapping from @xmath296 onto a convex neighborhood @xmath297 of @xmath298 . under the mapping @xmath295 , we then have that @xmath299 corresponds to @xmath300 and that @xmath301 corresponds to @xmath302 \cap f(u_{\delta})$ ] . we can also deform @xmath240 in such a way that for an arbitrary , but fixed @xmath303 , @xmath153 maps the portion of @xmath304 and @xmath305 in @xmath306 to the rays @xmath307 and @xmath308 , respectively .    with this mapping we take @xmath309 , analytic for @xmath310 , and continuous on @xmath311 , given by @xmath312 where @xmath313 and @xmath314 is an explicit matrix valued function built out of the airy function @xmath315 and its derivative @xmath316 as follows @xmath317      \left(\begin{array}{cc }      \operatorname{ai}(t ) & \operatorname{ai}(\omega^2 t ) \\ \operatorname{ai}'(t ) & \omega^2 \operatorname{ai}'(\omega^2 t ) \end{array } \right )      e^{- \frac{\pi i}{6 } \sigma_3 }      \left(\begin{array}{cc } 1 & 0 \\ -1 & 1 \end{array } \right ) &      \mbox{for } \theta < \arg t < \pi , \\[10pt ]      \left(\begin{array}{cc }      \operatorname{ai}(t ) & -\omega^2 \operatorname{ai}(\omega t ) \\ \operatorname{ai}'(t ) & -\operatorname{ai}'(\omega t ) \end{array } \right )      e^{-\frac{\pi i}{6 } \sigma_3 }      \left(\begin{array}{cc } 1 & 0 \\ 1 & 1 \end{array } \right ) &      \mbox{for } -\pi < \arg t < -\theta , \\[10pt ]      \left(\begin{array}{cc }      \operatorname{ai}(t ) & -\omega^2 \operatorname{ai}(\omega t ) \\ \operatorname{ai}'(t ) & -\operatorname{ai}'(\omega t ) \end{array } \right )      e^{- \frac{\pi i}{6 } \sigma_3 } &      \mbox{for } -\theta < \arg t < 0 ,      \end{array } \right.\ ] ] with @xmath318 .    a similar construction yields a parametrix @xmath319 in a neighborhood @xmath320 , see @xcite for details .    finally , using @xmath284 , @xmath309 , and @xmath319 , we define for every @xmath321 , @xmath322    then @xmath29 is defined and analytic on @xmath323 . however it follows from the construction that @xmath29 has no jumps on @xmath75 and on @xmath324 . therefore @xmath29 has an analytic continuation to @xmath325 , where @xmath326 is the contour indicated in figure [ fig : lens2 ] . contour @xmath326 splits the complex plane in the subdomains also indicated in figure [ fig : lens2 ] . lens2 ( 20,19)@xmath210 ( -4,35)@xmath249 ( 20.5,35)@xmath250 ( 60,55)@xmath251 ( 35,54)@xmath296 ( 8,14)@xmath327 ( 51,7)@xmath252 ( 90,55)@xmath328 ( 15,42)@xmath329 ( 50,40)@xmath330    we also have that @xmath331 with a constant @xmath216 that is independent of @xmath139 ( it can also be chosen independently of the value of @xmath6 for @xmath6 in a compact subset of @xmath106 , see @xcite ) , and standard arguments show that jump matrices for @xmath29 are close to the identity matrix if @xmath2 is large . although in the previous analysis we have used the value @xmath6 ( assumed fixed ) , as it was explained in the introduction to theorem  [ thm : stronga ] , all of the notions and results introduced before are @xmath2-dependent . however , from the asymptotic assumption we have that the curves @xmath193 tend to the limiting curve @xmath75 , etc . we observed already that the jump matrix for @xmath332 is @xmath333 uniformly on @xmath334 as @xmath196 . in addition , the jump matrix converges to the identity matrix as @xmath176 along the unbounded components of @xmath334 sufficiently fast , so that the jump matrix is also close to @xmath335 in the @xmath336-sense . since the contours @xmath334 are only slightly varying with @xmath2 , we may follow standard arguments to conclude that @xmath337 uniformly for @xmath338 .    finally , unraveling the steps @xmath339 and using ( [ eq72 ] ) , we obtain strong asymptotics for @xmath340 in all regions of the complex plane . in particular we are interested in the ( 1,1 ) entry of @xmath340 , since this is the monic generalized laguerre polynomial . we are not describing the details of this straightforward calculation here , and refer again the interested reader to @xcite . this completes the proof of theorem [ thm : stronga ] . we finish by proving corollary [ corollary : zeros ] . the assertion that all zeros of @xmath28 are in @xmath204 ( accumulate at @xmath75 ) is a direct consequence of ( a ) of theorem [ thm : stronga ] and the fact that @xmath341 in @xmath342 . observe that @xmath343 if and only if @xmath344 $ ] , where as in lemma  [ lemma : convexity ] , we denote by @xmath345 $ ] the straight segment joining @xmath80 and @xmath98 .    by lemma  [ lemma : convexity ] , for @xmath346 , @xmath347 $ ] is a boundary of a simply connected domain ; let us denote it by @xmath348 . thus , @xmath349 preserves sign both in @xmath348 and in @xmath350 , and these signs are opposite . since @xmath351 we conclude that @xmath352 for @xmath353 , and @xmath354 for @xmath355 . in particular , @xmath356 , the inner boundary of @xmath348 corresponds to the `` @xmath77 '' side of @xmath75 . since for @xmath346 , @xmath357 $ ] , we conclude that this property holds for every value of @xmath346 . in particular , we have proved that holds only on the `` @xmath77 '' side of @xmath75 .    by assertions ( b ) and ( c ) of theorem [ thm : stronga ] , zeros of @xmath28 in a neighborhood of @xmath193 must satisfy @xmath358since by , @xmath359 on both sides of @xmath131 , it remains to use to conclude the proof . the second and the third authors ( amf and pmg ) have been supported in part by the research project mtm2011 - 28952-c02 - 01 from the ministry of science and innovation of spain and the european regional development fund ( erdf ) , by junta de andaluca , research group fqm-229 , and by campus de excelencia internacional del mar ( ceimar ) of the university of almera . additionally , amf was supported by the excellence grant p09-fqm-4643 from junta de andaluca . mja and ft have been partially supported by the research unit ur11es87 from the university of gabs and the ministry of higher education and scientific research in tunisia . the authors acknowledge the contribution of the anonymous referee whose careful 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in this paper we study the asymptotics ( as @xmath0 ) of the sequences of laguerre polynomials with varying complex parameters @xmath1 depending on the degree @xmath2 . 
 more precisely , we assume that @xmath3 and @xmath4 . 
 this study has been carried out previously only for @xmath5 , but complex values of @xmath6 introduce an asymmetry that makes the problem more difficult . 
 the main ingredient of the asymptotic analysis is the right choice of the contour of orthogonality , which requires the analysis of the global structure of trajectories of an associated quadratic differential on the complex plane , which may have an independent interest . 
 while the weak asymptotics is obtained by reduction to the theorem of gonchar  rakhmanov  stahl , the strong asymptotic results are derived via the non - commutative steepest descent analysis based on the riemann - hilbert characterization of the laguerre polynomials . 
 trajectories and orthogonal trajectories of a quadratic differential , riemann - hilbert problems , generalized laguerre polynomials , strong and weak asymptotics , logarithmic potential , equilibrium .