in mathematics , katugampola fractional operators are integral operators that generalize the riemann – liouville and the hadamard fractional operators into a unique form .

the katugampola fractional integral generalizes both the riemann – liouville fractional integral and the hadamard fractional integral into a single form and it is also closely related to the erdelyi – kober operator that generalizes the riemann – liouville fractional integral .

katugampola fractional derivative has been defined using the katugampola fractional integral and as with any other fractional differential operator , it also extends the possibility of taking real number powers or complex number powers of the integral and differential operators .

katugampola fractional operators

katugampola , un : new approach to generalized fractional integral . appl . math . comput .
this manuscript is based on the standard fractional calculus iteration procedure on conformable derivatives . we introduce new fractional integration and differentiation operators . we define spaces and present some theorems related to these operators .
hua chen , and udita n . katugampola , “hermite–hadamard and hermite–hadamard–fejér type inequalities for generalized fractional integrals , ”
katugampola , un : a new approach to generalized fractional derivatives . bull . math . anal . appl .
giorgio s . taverna , and delfim f . m . torres , “generalized fractional operators for nonstandard lagrangians , ”
in this paper we present a generalization to two existing fractional integrals and derivatives , namely , the riemann - liouville and hadamard fractional operators . the existence and uniqueness results for single term fractional differential equations ( fde ) have also been established . we also obtain the mellin transforms of such generalized fractional operators which are important in solving fractional differential equations .
abdeljawad , t , baleanu , d : monotonicity results for fractional difference operators with discrete exponential kernels . adv . differ . equ .
ryan m . evans , udita n . katugampola , and david a . edwards , “applications of fractional calculus in solving abel - type integral equations : surface–volume reaction problem , ”
this paper was devoted to an investigation of the fractional derivatives and integrals obtained by iterating conformable integrals . we obtained left - and right - fractional conformable integrals . with a standard fractional procedure we found left - and right - fractional conformable derivatives in the sense of riemann - liouville and caputo . we proved that these fractional integrals and derivatives have properties similar to the standard fractional integrals and derivatives . we also define the fractional derivatives of functions belonging to specific spaces in order to find the relation between these new fractional differential operators . the presented left - and right - fractional integrals are different from those defined by katugampola since their kernels depend on the end points a and b and hence need a different convolution theory when the conformable laplace is applied .
gao , f , yang , xj : fractional maxwell fluid with fractional derivative without singular kernel . therm . sci .
respectively . we suggest that iterating the above integral will end up with new fractional operators with two parameters and kernels different from the usual kernels of usual fractional derivatives and integrals . from the data analysis point of view we suppose that this new type of calculus will provide better understanding of the complexity of the dynamics of the phenomena from porous media .
] . as is well known , systems possessing a memory effect often appear in real world phenomena . however , for each type of data we always ask what is the optimal corresponding nonlocal model to be applied . moreover , many authors studied new fractional operators with local , nonlocal , singular and non - singular kernels ( see [
for the right - fractional conformable derivatives , we can state a similar lemma .
podlubny , i : fractional differential equations . academic press , san diego ( 1999 )
kilbas , aa : hadamard type fractional calculus . j . korean math . soc .
, we define the left - and right - fractional conformable integrals and derivatives . in section
magin , rl : fractional calculus in bioengineering . begell house publishers , redding ( 2006 )
abdeljawad , t : on conformable fractional calculus . j . comput . appl . math .
we study incommensurate fractional variational problems in terms of a generalized fractional integral with lagrangians depending on classical derivatives and generalized fractional integrals and derivatives . we obtain necessary optimality conditions for the basic and isoperimetric problems , transversality conditions for free boundary value problems , and a generalized noether type theorem .
] , in what follows , we recall some basic definitions and tools about classical fractional calculus .
hilfer , r : applications of fractional calculus in physics . word scientific , singapore ( 2000 )
, we define the fractional conformable derivatives of functions belonging to certain spaces and state their properties . in section
matheus j . lazo , and delfim f . m . torres , “variational calculus with conformable fractional derivatives , ”
in this section , we consider the fractional conformable derivatives of functions belonging to spaces stated in the following definitions .
] and the references therein ) . the standard fractional calculus may not provide us the required kernel in order to extract important information from these types of systems . at this stage , we ask the following question . can we generalize the standard fractional riemann - liouville integrals in a way such that we obtain unification to riemann - liouville , hadamard and other fractional derivatives [
before we state the inverse properties and the composition of two caputo fractional conformable derivatives , we shall consider the following lemmas .
el - nabulsi , ra , torres , dfm : fractional action - like variational problems . j . math . phys .
in this section we define the left - and right - fractional conformable derivatives in the sense of caputo and present their properties .
a . b . malinowska , and d . f . m . torres , “fractional calculus of variations of several independent variables , ”
shakoor pooseh , ricardo almeida , and delfim f . m . torres , “discrete direct methods in the fractional calculus of variations , ”
dina tavares , ricardo almeida , and delfim f . m . torres , “caputo derivatives of fractional variable order : numerical approximations , ”
atangana , a , baleanu , d : new fractional derivative with non - local and non - singular kernel . therm . sci .
a generalized fractional calculus of variations - control and cybernetics - tom vol . 42 , no . 2 ( 2013 ) - baztech - yadda
tatiana odzijewicz , agnieszka b . malinowska , and delfim f . m . torres , “variable order fractional variational calculus for double integrals , ”
tatiana odzijewicz , agnieszka b . malinowska , and delfim f . m . torres , “green ' s theorem for generalized fractional derivatives , ”
michael a . zhuravkov , and natalie s . romanova , “review of methods and approaches for mechanical problem solutions based on fractional calculus , ”
roberto garra , giorgio s . taverna , and delfim f . m . torres , “fractional herglotz variational principles with generalized caputo derivatives , ”
we present the fractional conformable derivatives in the caputo setting and state their properties . finally , the last section is devoted to our conclusion .
caputo , m , fabrizio , m : a new definition of fractional derivative without singular kernel . prog . fract . differ . appl .
losada , j , nieto , jj : properties of a new fractional derivative without singular kernel . prog . fract . differ . appl .
in the area of fractional calculus and its applications in many branches of science and engineering , several fractional derivatives were mainly utilized . the most common used were caputo and riemann - liouville derivatives , which were successfully utilized in modeling complex dynamics appearing in physics , biology , engineering and many other fields [
jarad , f , abdeljawad , t , baleanu , d : caputo - type modification of the hadamard fractional derivative . adv . differ . equ .
gastao s . f . frederico , and delfim f . m . torres , “fractional noether ' s theorem with classical and riemann - liouville derivatives , ”
samko , sg , kilbas , aa , marichev , oi : fractional integrals and derivatives : theory and applications . gordon & breach , yverdon ( 1993 )
yang , xj , gao , f , machado , jat , baleanu , d : a new fractional derivative involving the normalized sinc function without singular kernel .
jarad , f , abdeljawad , t , baleanu , d : on the generalized fractional derivatives and their caputo modification . j . nonlinear sci . appl .
mohammad maleki , ishak hashim , saeid abbasbandy , and a . alsaedi , “direct solution of a type of constrained fractional variational problems via an adaptive pseudospectral method , ”
abdeljawad , t , baleanu , d : on fractional derivatives with exponential kernel and their discrete versions . j . rep . math . phys . ( to appear )
gambo , yy , jarad , f , abdeljawad , t , baleanu , d : on caputo modification of the hadamard fractional derivative . adv . differ . equ .
kilbas , a , srivastava , hm , trujillo , jj : theory and application of fractional differential equations . north - holland mathematics studies , vol . 204 ( 2006 )
hegagi m . ali , fernando lobo pereira , and sãlvio m . a . gama , “a new approach to the pontryagin maximum principle for nonlinear fractional optimal control problems , ”
adjabi , y , jarad , f , baleanu , d , abdeljawad , t : on cauchy problems with caputo hadamard fractional derivatives . j . comput . anal . appl .
matheus j . lazo , and delfim f . m . torres , “the dubois–reymond fundamental lemma of the fractional calculus of variations and an euler–lagrange equation involving only derivatives of caputo , ”
loic bourdin , tatiana odzijewicz , and delfim f . m . torres , “existence of minimizers for generalized lagrangian functionals and a necessary optimality condition - application to fractional variational problems , ”
abdeljawad , t , baleanu , d : integration by parts and its applications of a new nonlocal fractional derivative with mittag - leffler nonsingular kernel . j . nonlinear sci . appl .
s . s . ezz - eldien , e . h . doha , a . h . bhrawy , a . a . el - kalaawy , and j . a . t . machado , “a new operational approach for solving fractional variational problems depending on indefinite integrals , ”
] . the core of this procedure is to decide which differentiation operator should be used as a starting point for the iteration procedure . for the standard fractional calculus , we iterate the usual integral of a function and using the cauchy formula we obtain the integral of higher integer orders and then replace this integer by any complex number . in [
this dissertation is only available for download to the siuc community . others should contact the interlibrary loan department of your local library or contact proquest ' s dissertation express service .
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faculty of computer science , bialystok university of technology , 15 - 351 białystok , poland , a . malinowska @ urltoken
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tatiana odzijewicz , 1 agnieszka b . malinowska , 2 and delfim f . m . torres 1
copyright © 2012 tatiana odzijewicz et al . this is an open access article distributed under the creative commons attribution license , which permits unrestricted use , distribution , and reproduction in any medium , provided the original work is properly cited .
the following is the list of published articles that have cited the current article .
delfim f . m . torres , and agnieszka b . malinowskapp . 1–275 , 2012 .
rajesh k . pandey , and om p . agrawal , “numerical scheme for generalized isoperimetric constraint variational problems with a - operator , ”
rajesh k . pandey , and om p . agrawal , “numerical scheme for a quadratic type generalized isoperimetric constraint variational problems with a - operator , ”
] , it was suggested that the conformable integral should be fractionalized properly . we recall that an integral type like the one from [
$ $ \ bigl ( \ mathbf { i } _ { b } ^ { \ alpha } f \ bigr ) ( t ) = \ frac { 1 } { \ gamma ( \ alpha ) } \ int _ { t } ^ { b } ( y - t ) ^ { \ alpha - 1 } f ( y ) \ , dy . $ $
$ $ \ bigl ( { } _ { a } d ^ { \ alpha } f \ bigr ) ( t ) = \ biggl ( \ frac { d } { dt } \ biggr ) ^ { n } \ bigl ( { } _ { a } \ mathbf { i } ^ { n - \ alpha } f \ bigr ) ( t ) , \ quad n = [ \ alpha ] + 1 . $ $
$ $ \ bigl ( d _ { b } ^ { \ alpha } f \ bigr ) ( t ) = \ biggl ( - \ frac { d } { dt } \ biggr ) ^ { n } \ bigl ( \ mathbf { i } _ { b } ^ { n - \ alpha } f \ bigr ) ( t ) . $ $
$ $ \ bigl ( { } ^ { c } d _ { b } ^ { \ alpha } f \ bigr ) ( t ) = \ bigl ( \ mathbf { i } _ { b } ^ { n - \ alpha } ( - 1 ) ^ { n } f ^ { ( n ) } \ bigr ) ( t ) . $ $
$ $ \ bigl ( \ mathfrak { j } _ { b } ^ { \ alpha } f \ bigr ) ( t ) = \ frac { 1 } { \ gamma ( \ alpha ) } \ int _ { t } ^ { b } ( \ ln y - \ ln t ) ^ { \ alpha - 1 } f ( y ) \ frac { dy } { y } . $ $
$ $ \ bigl ( \ mathfrak { d } _ { b } ^ { \ alpha } f \ bigr ) ( t ) = \ biggl ( - t \ frac { d } { dt } \ biggr ) ^ { n } \ bigl ( i _ { b } ^ { n - \ alpha } f \ bigr ) ( t ) . $ $
$ $ \ bigl ( { } ^ { c } \ mathfrak { d } _ { b } ^ { \ alpha } f \ bigr ) ( t ) = \ biggl ( \ mathfrak { j } _ { b } ^ { n - \ alpha } \ biggl ( - t \ frac { d } { dt } \ biggr ) ^ { n } f \ biggr ) ( t ) . $ $
$ $ x _ { c } ^ { p } ( a , b ) = \ biggl \ { f : [ a , b ] \ rightarrow \ mathbb { r } : \ | f \ | _ { x _ { c } ^ { p } } = \ biggl ( \ int _ { a } ^ { b } \ bigl \ vert t ^ { c } f ( t ) \ bigr \ vert ^ { p } \ frac { dt } { t } \ biggr ) ^ { 1 / p } < \ infty \ biggr \ } . $ $
$ $ \ bigl ( { } ^ { c } d _ { b } ^ { \ alpha , \ rho } f \ bigr ) ( t ) = \ bigl ( { } _ { a } i ^ { n - \ alpha , \ rho } ( - \ gamma ) ^ { n } f \ bigr ) ( t ) = \ frac { 1 } { \ gamma ( n - \ alpha ) } \ int _ { t } ^ { b } \ biggl ( \ frac { y ^ { \ rho } - t ^ { \ rho } } { \ rho } \ biggr ) ^ { n - \ alpha - 1 } ( - \ gamma ) ^ { n } f ( y ) \ frac { dy } { y ^ { 1 - \ rho } } . $ $
$ $ \ begin { aligned } \ begin { aligned } [ b ] { } _ { a } i ^ { n , \ alpha } f ( x ) & = \ int _ { a } ^ { x } \ frac { dt _ { 1 } } { ( t _ { 1 } - a ) ^ { 1 - \ alpha } } \ int _ { a } ^ { t _ { 1 } } \ frac { dt _ { 2 } } { ( t _ { 2 } - a ) ^ { 1 - \ alpha } } \ cdots \ int _ { a } ^ { t _ { n - 1 } } \ frac { f ( t _ { n } ) \ , dt _ { n } } { ( t _ { n } - a ) ^ { 1 - \ alpha } } \ \ & = \ frac { 1 } { \ gamma ( n ) } \ int _ { a } ^ { x } \ biggl ( \ frac { ( x - a ) ^ { \ alpha } - ( t - a ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { n - 1 } f ( t ) \ frac { dt } { ( t - a ) ^ { 1 - \ alpha } } . \ end { aligned } \ end { aligned } $ $
$ $ { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f ( x ) = \ frac { 1 } { \ gamma ( \ beta ) } \ int _ { a } ^ { x } \ biggl ( \ frac { ( x - a ) ^ { \ alpha } - ( t - a ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { \ beta - 1 } f ( t ) \ frac { dt } { ( t - a ) ^ { 1 - \ alpha } } . $ $
$ $ { } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } f ( x ) = \ frac { 1 } { \ gamma ( \ beta ) } \ int _ { x } ^ { b } \ biggl ( \ frac { ( b - x ) ^ { \ alpha } - ( b - t ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { \ beta - 1 } f ( t ) \ frac { dt } { ( b - t ) ^ { 1 - \ alpha } } . $ $
$ $ { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } \ bigl ( { } _ { a } ^ { \ gamma } \ mathfrak { i } ^ { \ alpha } \ bigr ) f ( x ) = { } _ { a } ^ { \ beta + \ gamma } \ mathfrak { i } ^ { \ alpha } f ( x ) , \ qquad { } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } \ bigl ( { } ^ { \ gamma } \ mathfrak { i } _ { b } ^ { \ alpha } \ bigr ) f ( x ) = { } ^ { \ beta + \ gamma } \ mathfrak { i } _ { b } ^ { \ alpha } f ( x ) . $ $
$ $ z = \ frac { ( t - a ) ^ { \ alpha } - ( u - a ) ^ { \ alpha } } { ( x - a ) ^ { \ alpha } - ( u - a ) ^ { \ alpha } } . $ $
the second formula can be proved in a similar way or by using the action of the q - operator . □
$ $ \ begin { aligned } & \ bigl ( { } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } ( b - t ) ^ { \ alpha \ nu - \ alpha } \ bigr ) ( x ) = \ frac { 1 } { \ alpha ^ { \ beta } } \ frac { \ gamma ( \ nu ) } { \ gamma ( \ beta + \ nu ) } ( b - x ) ^ { \ alpha ( \ beta + \ nu - 1 ) } . \ end { aligned } $ $
$ $ \ bigl ( { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } ( t - a ) ^ { \ nu - 1 } \ bigr ) ( x ) = \ frac { 1 } { \ gamma ( \ beta ) } \ int _ { a } ^ { x } \ biggl ( \ frac { ( x - a ) ^ { \ alpha } - ( t - a ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { \ beta - 1 } ( t - a ) ^ { \ alpha \ nu - \ alpha } \ frac { dt } { ( t - a ) ^ { 1 - \ alpha } } . $ $
$ $ \ bigl ( { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } ( t - a ) ^ { \ nu - 1 } \ bigr ) ( x ) = \ frac { ( x - a ) ^ { \ alpha ( \ beta + \ nu - 1 ) } } { \ gamma ( \ beta ) \ alpha ^ { \ beta } } \ int _ { 0 } ^ { 1 } ( 1 - u ) ^ { \ beta - 1 } u ^ { \ nu - 1 } \ , du = \ frac { \ gamma ( \ nu ) } { \ alpha ^ { \ beta } \ gamma ( \ beta + \ nu ) } ( x - a ) ^ { \ alpha ( \ beta + \ nu - 1 ) } . $ $
$ $ \ begin { aligned } & \ bigl [ { } ^ { \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } ( b - t ) ^ { \ alpha \ nu - \ alpha } \ bigr ] ( x ) = \ alpha ^ { \ beta } \ frac { \ gamma ( \ nu ) } { \ gamma ( \ nu - \ beta ) } ( b - x ) ^ { \ alpha ( \ nu - \ beta - 1 ) } . \ end { aligned } $ $
$ $ { } _ { a } ^ { \ beta } \ mathfrak { d } ^ { \ alpha } f = { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { - \ alpha } f , \ qquad { } ^ { \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } = { } ^ { \ beta } \ mathfrak { i } _ { b } ^ { - \ alpha } f . $ $
$ $ \ frac { d } { dx } \ bigl ( { } _ { a } ^ { n - 2 } t ^ { \ alpha } f ( x ) \ bigr ) = \ frac { 1 } { ( x - a ) ^ { 1 - \ alpha } } \ int _ { a } ^ { x } \ psi ( t ) \ frac { dt } { ( t - a ) ^ { 1 - \ alpha } } + \ frac { { } _ { a } ^ { n - 1 } t ^ { \ alpha } f ( a ) } { ( x - a ) ^ { 1 - \ alpha } } . $ $
$ $ \ begin { aligned } { } _ { a } ^ { n - 2 } t ^ { \ alpha } f ( x ) = & \ int _ { a } ^ { x } \ frac { ( x - a ) ^ { \ alpha } - ( t - a ) ^ { \ alpha } } { \ alpha } \ psi ( t ) \ frac { dt } { ( t - a ) ^ { 1 - \ alpha } } \ \ & { } + { } _ { a } ^ { n - 1 } t ^ { \ alpha } f ( a ) \ frac { ( x - a ) ^ { \ alpha } } { \ alpha } + { } _ { a } ^ { n - 2 } t ^ { \ alpha } f ( a ) . \ end { aligned } $ $
$ $ \ begin { aligned } { } _ { a } ^ { n - 3 } t ^ { \ alpha } f ( x ) = & \ frac { 1 } { 2 ! } \ int _ { a } ^ { x } \ biggl ( \ frac { ( x - a ) ^ { \ alpha } - ( t - a ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { 2 } \ psi ( t ) \ frac { dt } { ( t - a ) ^ { 1 - \ alpha } } + { } _ { a } ^ { n - 2 } t ^ { \ alpha } f ( a ) \ frac { ( x - a ) ^ { 2 \ alpha } } { 2 \ alpha ^ { 2 } } \ \ & { } + { } _ { a } ^ { n - 2 } t ^ { \ alpha } f ( a ) \ frac { ( x - a ) ^ { \ alpha } } { \ alpha } + { } _ { a } ^ { n - 3 } t ^ { \ alpha } f ( a ) . \ end { aligned } $ $
$ $ \ begin { aligned } f ( x ) = & \ frac { 1 } { ( n - 1 ) ! } \ int _ { a } ^ { x } \ biggl ( \ frac { ( x - a ) ^ { \ alpha } - ( t - a ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { n - 1 } \ psi ( t ) \ frac { dt } { ( t - a ) ^ { 1 - \ alpha } } \ \ & { } + \ sum _ { k = 0 } ^ { n - 1 } \ frac { { } _ { a } ^ { k } t ^ { \ alpha } f ( a ) } { \ alpha ^ { k } k ! } ( x - a ) ^ { \ alpha k } . \ end { aligned } $ $
$ $ \ begin { aligned } f ( x ) = & \ frac { 1 } { ( n - 1 ) ! } \ int _ { x } ^ { b } \ biggl ( \ frac { ( b - x ) ^ { \ alpha } - ( b - t ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { n - 1 } \ frac { ( { } ^ { n } t _ { b } ^ { \ alpha } f ) ( t ) } { ( b - t ) ^ { 1 - \ alpha } } \ , dt \ \ & { } + \ sum _ { k = 0 } ^ { n - 1 } \ frac { ( - 1 ) ^ { k } \ , { } ^ { k } t _ { b } ^ { \ alpha } f ( b ) } { k ! } \ frac { ( b - x ) ^ { \ alpha k } } { \ alpha ^ { k } } . \ end { aligned } $ $
$ $ \ begin { aligned } { } ^ { \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } f ( x ) = & \ frac { 1 } { \ gamma ( n - \ beta ) } \ int _ { x } ^ { b } \ biggl ( \ frac { ( b - x ) ^ { \ alpha } - ( b - t ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { n - \ beta - 1 } \ frac { ^ { n } t _ { b } ^ { \ alpha } f ( t ) } { ( b - t ) ^ { 1 - \ alpha } } \ , dt \ \ & { } + \ sum _ { k = 0 } ^ { n - 1 } \ frac { ( ( - 1 ) ^ { k } \ , { } ^ { k } t _ { b } ^ { \ alpha } f ( b ) ) ( b - x ) ^ { \ alpha ( k - \ beta ) } } { \ alpha ^ { k - \ beta } \ gamma ( k - \ beta + 1 ) } . \ end { aligned } $ $
$ $ f ( x ) = \ frac { 1 } { ( n - 1 ) ! } \ int _ { a } ^ { x } \ biggl ( \ frac { ( x - a ) ^ { \ alpha } - ( t - a ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { n - 1 } \ frac { ( { } _ { a } ^ { n } t ^ { \ alpha } f ) ( t ) } { ( t - a ) ^ { 1 - \ alpha } } \ , dt + \ sum _ { k = 0 } ^ { n - 1 } \ frac { { } _ { a } ^ { n } t ^ { \ alpha } f ( a ) } { k ! } \ frac { ( x - a ) ^ { \ alpha k } } { \ alpha ^ { k } } . $ $
$ $ \ begin { aligned } { } _ { a } ^ { \ beta } \ mathfrak { d } ^ { \ alpha } f ( x ) = & \ frac { { } _ { a } ^ { n } t ^ { \ alpha } } { ( n - 1 ) ! \ gamma ( n - \ beta ) } \ int _ { a } ^ { x } \ int _ { a } ^ { t } \ biggl ( \ frac { ( x - a ) ^ { \ alpha } - ( t - a ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { n - \ beta - 1 } \ biggl ( \ frac { ( t - a ) ^ { \ alpha } - ( u - a ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { n } \ \ & { } \ times \ bigl ( { } _ { a } ^ { n } t ^ { \ alpha } f ( u ) \ bigr ) \ frac { du } { ( u - a ) ^ { 1 - \ alpha } } \ frac { dt } { ( t - a ) ^ { 1 - \ alpha } } \ \ & { } + \ sum _ { k = 0 } ^ { n - 1 } \ frac { { } _ { a } ^ { n } t ^ { \ alpha } f ( a ) } { k ! \ alpha ^ { k } } \ alpha ^ { \ beta } \ frac { \ gamma ( k + 1 ) } { \ gamma ( k + 1 - \ beta ) } ( x - a ) ^ { \ alpha ( k - \ beta ) } . \ end { aligned } $ $
$ $ \ begin { aligned } { } _ { a } ^ { \ beta } \ mathfrak { d } ^ { \ alpha } f ( x ) = & \ biggl ( \ frac { \ gamma ( n - \ beta ) \ gamma ( n ) { } _ { a } ^ { n } t ^ { \ alpha } } { ( n - 1 ) ! \ gamma ( n - \ beta ) \ gamma ( 2n - \ beta ) } \ int _ { a } ^ { x } \ biggl ( \ frac { ( x - a ) ^ { \ alpha } - ( u - a ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { 2n - \ beta - 1 } \ frac { ( { } _ { a } ^ { n } t ^ { \ alpha } f ) ( u ) \ , du } { ( u - a ) ^ { 1 - \ alpha } } \ biggr ) \ \ & { } + \ sum _ { k = 0 } ^ { n - 1 } \ frac { { } _ { a } ^ { n } t ^ { \ alpha } f ( a ) } { \ alpha ^ { k - \ beta } \ gamma ( k + 1 - \ beta ) } ( x - a ) ^ { \ alpha ( k - \ beta ) } . \ end { aligned } $ $
$ $ { } _ { a } ^ { m } t ^ { \ alpha } \ bigl ( { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f ( x ) \ bigr ) = { } _ { a } ^ { \ beta - m } \ mathfrak { i } ^ { \ alpha } f ( x ) ; \ qquad { } ^ { m } t _ { b } ^ { \ alpha } \ bigl ( { } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } f ( x ) \ bigr ) = { } ^ { \ beta - m } \ mathfrak { i } _ { b } ^ { \ alpha } f ( x ) . $ $
$ $ { } _ { a } ^ { \ gamma } \ mathfrak { d } ^ { \ alpha } \ bigl ( { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f ( x ) \ bigr ) = { } _ { a } ^ { \ beta - \ gamma } \ mathfrak { i } ^ { \ alpha } f ( x ) ; \ qquad { } ^ { \ gamma } \ mathfrak { d } _ { b } ^ { \ alpha } \ bigl ( { } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } f ( x ) \ bigr ) = ^ { \ beta - \ gamma } \ mathfrak { i } _ { b } ^ { \ alpha } f ( x ) . $ $
$ $ \ begin { aligned } { } _ { a } ^ { \ gamma } \ mathfrak { d } ^ { \ alpha } \ bigl ( { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f ( x ) \ bigr ) & = { } _ { a } ^ { m } t ^ { \ alpha } \ bigl ( { } _ { a } ^ { m - \ gamma } \ mathfrak { i } ^ { \ alpha } \ bigl ( { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f ( x ) \ bigr ) \ bigr ) \ \ & = { } _ { a } ^ { m } t ^ { \ alpha } \ bigl ( { } _ { a } ^ { \ beta + m - \ gamma } \ mathfrak { i } ^ { \ alpha } f ( x ) \ bigr ) \ \ & = { } _ { a } ^ { \ beta - \ gamma } \ mathfrak { i } ^ { \ alpha } f ( x ) . \ end { aligned } $ $
$ $ { } _ { a } ^ { \ beta } \ mathfrak { d } ^ { \ alpha } \ bigl ( { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f ( x ) \ bigr ) = f ( x ) ; \ qquad { } ^ { \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } \ bigl ( { } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } f ( x ) \ bigr ) = f ( x ) . $ $
and utilized the properties of the gamma and beta functions . the last step in the proof is to use lemma 2 . 1 in [
$ $ { } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } \ bigl ( { } ^ { \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } f ( x ) \ bigr ) = f ( x ) - \ sum _ { j = 1 } ^ { n } \ frac { ( - 1 ) ^ { n - j } \ , { } ^ { \ beta - j } \ mathfrak { d } _ { b } ^ { \ alpha } f ( b ) } { \ alpha ^ { \ beta - j } \ gamma ( \ beta - j + 1 ) } ( b - x ) ^ { \ alpha \ beta - \ alpha j } . $ $
$ $ \ begin { aligned } { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } \ bigl ( { } _ { a } ^ { \ beta } \ mathfrak { d } ^ { \ alpha } f ( x ) \ bigr ) = & \ frac { 1 } { \ gamma ( \ beta ) } \ int _ { a } ^ { x } \ biggl ( \ frac { ( x - a ) ^ { \ alpha } - ( t - a ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { \ beta - 1 } \ bigl ( { } _ { a } ^ { n } t ^ { \ alpha } \ bigl ( { } _ { a } ^ { n - \ beta } \ mathfrak { i } ^ { \ alpha } f ( t ) \ bigr ) \ bigr ) \ frac { dt } { ( t - a ) ^ { 1 - \ alpha } } \ \ = & \ frac { { } _ { a } ^ { 1 } t ^ { \ alpha } } { \ gamma ( \ beta + 1 ) } \ biggl [ \ int _ { a } ^ { x } \ biggl ( \ frac { ( x - a ) ^ { \ alpha } - ( t - a ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { \ beta } \ bigl ( { } _ { a } ^ { n } t ^ { \ alpha } \ bigl ( { } _ { a } ^ { n - \ beta } \ mathfrak { i } ^ { \ alpha } f ( t ) \ bigr ) \ bigr ) \ frac { dt } { ( t - a ) ^ { 1 - \ alpha } } \ biggr ] . \ end { aligned } $ $
$ $ \ begin { aligned } { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } \ bigl ( { } _ { a } ^ { \ beta } \ mathfrak { d } ^ { \ alpha } f ( x ) \ bigr ) = & { } _ { a } ^ { 1 } t \ biggl [ { } _ { a } ^ { 1 } \ mathfrak { i } ^ { \ alpha } f ( x ) - \ sum _ { j = 1 } ^ { n } \ frac { ( { } _ { a } ^ { n - j } t ^ { \ alpha } ( { } _ { a } ^ { n - \ beta } \ mathfrak { i } ^ { \ alpha } f ( a ) ) ) } { \ gamma ( \ beta + 2 - j ) \ alpha ^ { \ beta - j + 1 } } ( x - a ) ^ { \ alpha \ beta - \ alpha j + \ alpha } \ biggr ] \ \ = & f ( x ) - \ sum _ { j = 1 } ^ { n } \ frac { { } _ { a } ^ { \ beta - j } \ mathfrak { d } ^ { \ alpha } f ( a ) } { \ alpha ^ { \ beta - j } \ gamma ( \ beta - j + 1 ) } ( x - a ) ^ { \ alpha \ beta - \ alpha j } . \ end { aligned } $ $
$ $ { } ^ { c \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } f ( x ) = { } ^ { \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } \ biggl [ f ( t ) - \ sum _ { k = 0 } ^ { n - 1 } \ frac { ( - 1 ) ^ { k } \ , { } ^ { k } t _ { b } ^ { \ alpha } f ( b ) } { k ! \ alpha ^ { k } } ( b - t ) ^ { \ alpha k } \ biggr ] ( x ) . $ $
$ $ \ begin { aligned } { } ^ { c \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } f ( x ) & = \ frac { ( - 1 ) ^ { n } } { \ gamma ( n - \ beta ) } \ int _ { x } ^ { b } \ biggl ( \ frac { ( b - x ) ^ { \ alpha } - ( b - t ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { n - \ beta - 1 } \ frac { ^ { n } t _ { b } ^ { \ alpha } f ( t ) } { ( b - t ) ^ { 1 - \ alpha } } \ , dt \ \ & = { } ^ { n - \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } \ bigl ( ^ { n } t _ { b } ^ { \ alpha } f ( x ) \ bigr ) . \ end { aligned } $ $
$ $ \ begin { aligned } { } ^ { c \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } f ( x ) & = { } _ { a } ^ { \ beta } \ mathfrak { d } ^ { \ alpha } f ( x ) - \ sum _ { k = 0 } ^ { n - 1 } \ frac { ^ { k } t _ { a } ^ { \ alpha } f ( a ) } { \ alpha ^ { k - \ beta } k ! } \ frac { \ gamma ( k + 1 ) } { \ gamma ( k - \ beta + 1 ) } ( x - a ) ^ { k \ alpha - \ beta \ alpha } \ \ & = { } _ { a } ^ { \ beta } \ mathfrak { d } ^ { \ alpha } f ( x ) - \ sum _ { k = 0 } ^ { n - 1 } \ frac { ^ { k } t _ { a } ^ { \ alpha } f ( a ) } { \ alpha ^ { k - \ beta } \ gamma ( k - \ beta + 1 ) } ( x - a ) ^ { k \ alpha - \ beta \ alpha } \ \ & = \ frac { 1 } { \ gamma ( n - \ beta ) } \ int _ { a } ^ { x } \ biggl ( \ frac { ( x - a ) ^ { \ alpha } - ( t - a ) ^ { \ alpha } } { \ rho } \ biggr ) ^ { n - \ beta - 1 } \ frac { { } _ { a } ^ { n } t ^ { \ alpha } f ( t ) } { ( t - a ) ^ { 1 - \ alpha } } \ , dt \ \ & = { } _ { a } ^ { n - \ beta } \ mathfrak { i } ^ { \ alpha } \ bigl ( { } _ { a } ^ { n } t ^ { \ alpha } f ( x ) \ bigr ) . \ end { aligned } $ $
$ $ \ bigl \ vert { } _ { a } ^ { \ beta - k } \ mathfrak { i } ^ { \ alpha } f ( x ) \ bigr \ vert \ leq \ frac { \ vert f \ vert _ { c } } { \ vert \ gamma ( \ beta - k ) \ vert ( \ operatorname { re } ( \ beta ) - k ) } \ frac { ( x - a ) ^ { \ alpha ( \ operatorname { re } ( \ beta ) - k ) } } { \ alpha ^ { \ operatorname { re } ( \ beta ) - k } } . $ $
$ $ \ bigl \ vert ^ { c \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } f ( x ) \ bigr \ vert \ leq \ frac { \ vert ^ { n } t _ { b } ^ { \ alpha } f \ vert _ { c } } { \ vert \ gamma ( n - \ beta ) \ vert ( n - \ operatorname { re } ( \ beta ) ) } \ frac { ( b - x ) ^ { \ alpha ( n - \ operatorname { re } ( \ beta ) ) } } { \ alpha ^ { n - \ operatorname { re } ( \ beta ) } } . $ $
$ $ { } _ { a } ^ { c \ beta } \ mathfrak { d } ^ { \ alpha } \ bigl ( { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f ( x ) \ bigr ) = f ( x ) ; \ qquad { } ^ { c \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } \ bigl ( { } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } f ( x ) \ bigr ) = f ( x ) . $ $
$ $ \ begin { aligned } & { } ^ { c \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } \ bigl ( { } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } f ( x ) \ bigr ) = f ( x ) - \ frac { { } ^ { \ beta + 1 - n } \ mathfrak { i } _ { b } ^ { \ alpha } f ( b ) } { \ alpha ^ { n - \ beta } \ gamma ( n - \ beta ) } ( b - x ) ^ { \ alpha n - \ alpha \ beta } . \ end { aligned } $ $
$ $ { } _ { a } ^ { c \ beta } \ mathfrak { d } ^ { \ alpha } \ bigl ( { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f ( x ) \ bigr ) = { } _ { a } ^ { \ beta } \ mathfrak { d } ^ { \ alpha } \ bigl ( { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f ( x ) \ bigr ) - \ sum _ { k = 0 } ^ { n - 1 } \ frac { { } _ { a } ^ { k } t ^ { \ alpha } ( { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f ( a ) ) ( x - a ) } { \ alpha ^ { k - \ beta } \ gamma ( k - \ beta + 1 ) } . $ $
$ $ { } _ { a } ^ { c \ beta } \ mathfrak { d } ^ { \ alpha } \ bigl ( { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f ( x ) \ bigr ) = f ( x ) - \ sum _ { k = 0 } ^ { n - 1 } \ frac { { } _ { a } ^ { \ beta - k } \ mathfrak { i } ^ { \ alpha } f ( a ) ( x - a ) ^ { \ alpha - k - \ beta \ alpha } } { \ alpha ^ { k - \ beta } \ gamma ( k - \ beta + 1 ) } . $ $
) is proved . the second identity can be proved by using the same arguments .
$ $ \ begin { aligned } & { } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } \ bigl ( ^ { c \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } f ( x ) \ bigr ) = f ( x ) - \ sum _ { k = 0 } ^ { n - 1 } \ frac { ( - 1 ) ^ { k } \ , { } ^ { k } t _ { b } ^ { \ alpha } f ( b ) ( b - x ) ^ { \ alpha k } } { \ alpha ^ { k } k ! } . \ end { aligned } $ $
$ $ \ begin { aligned } { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } \ bigl ( { } _ { a } ^ { c \ beta } \ mathfrak { d } ^ { \ alpha } f ( x ) \ bigr ) & = { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } \ bigl ( { } _ { a } ^ { n - \ beta } \ mathfrak { i } ^ { \ alpha } \ bigl ( { } _ { a } ^ { n } t ^ { \ alpha } f ( x ) \ bigr ) \ bigr ) \ \ & = { } _ { a } ^ { n } \ mathfrak { i } ^ { \ alpha } \ bigl ( { } _ { a } ^ { n } t ^ { \ alpha } f ( x ) \ bigr ) \ \ & = f ( x ) - \ sum _ { j = 1 } ^ { n } \ frac { { } _ { a } ^ { n - j } d ^ { \ alpha } f ( a ) } { \ alpha ^ { n - j } \ gamma ( n - j + 1 ) } ( x - a ) ^ { ( n - j ) \ alpha } \ \ & = f ( x ) - \ sum _ { k = 0 } ^ { n - 1 } \ frac { { } _ { a } ^ { n } t ^ { \ alpha } f ( a ) } { \ alpha ^ { k } k ! } ( x - a ) ^ { k \ alpha } . \ end { aligned } $ $
$ $ { } _ { a } ^ { c \ beta } \ mathfrak { d } ^ { \ alpha } \ bigl ( { } _ { a } ^ { c \ gamma } \ mathfrak { d } ^ { \ alpha } f ( x ) \ bigr ) = { } _ { a } ^ { c ( \ beta + \ gamma ) } \ mathfrak { d } ^ { \ alpha } f ( x ) ; \ qquad { } ^ { c \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } \ bigl ( ^ { c \ gamma } \ mathfrak { d } _ { b } ^ { \ alpha } f ( x ) \ bigr ) = ^ { c ( \ beta + \ gamma ) } \ mathfrak { d } _ { b } ^ { \ alpha } f ( x ) . $ $
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the free state brewing company is a microbrewery based in lawrence , kansas .

it opened in 1989 as the first legal brewery in kansas in over 100 years .

in 2010 , the brewery announced plans to expand to distribute to vendors in kansas and missouri , and as far east as st. louis .

on may 21 , 2010 free state brewery began to distribute bottled beer to stores in lawrence , kansas .

four bottled beers were released : copperhead pale ale , oatmeal stout , ad astra ale , and wheat state golden .

free state brewing company

becca k . is drinking a cloud hopper imperial ipa by free state brewing co . at free state brewing company
dayle s . is drinking a free state golden by free state brewing co . at old chicago pizza & taproom – hays
free state brewing co . - lawrence , ks - beers and ratings | untappd
jeff is drinking a juicebox ipa by free state brewing co . at the anchor
chris k . is drinking an oatmeal stout by free state brewing co . at the hon room
the worst thing about the free state brewing co . is the distance between theri door and mine .
matt h . is drinking a copperhead pale ale by free state brewing co . at la casa de campsite
passed the institute for brewing and distilling’s general certification in brewing with distinction in 2008 .
when it comes to his love for beer , chuck magerl comes by it honestly . you see , many years before chuck founded free state brewing company , his grandfather was getting into the business himself – an act that was highly illegal during the height of prohibition , and leading to a stint in leavenworth .
in 1989 , free state brewing company became the first legal brewery in kansas in over 100 years . since opening the doors at our mass street hq , we’ve been working day in and day out to be a haven for beer lovers , where good beer and good food can inspire lively conversations and memorable experiences .
prepared with care using only the freshest ingredients , our daily specials make the perfect pairings with our tasty free state brews .
free state is always a stop when i visit lawrence . the food , the beer , and the service are always fantastic !
i’m a ‘92 jayhawk alum . brought my family to lawrence for the day . we chose free state brewing for lunch and was perfect . watched the world cup . great food . yummy ipa beer . good prices . super swee
is a reminder of my days at ku . always worth finding an excuse to get back to larry and have a little free state .
skip the wait stopping by the brew pub ? text “free state” to 626 - 414 - 3210 to get your name on the wait list .
want to learn more about the fellas that brew up your favorite free state beer ? of course you do , so come on in and read their stories .
but don’t worry – free state wasn’t founded on passion alone . chuck ' s background in biology , engineering and history is the foundation of the brewery , and he remains an avid student of all things beer .
a full - bodied stout with a well - rounded , balanced flavor and silky texture . this rich , tasty brew goes down smoothly , and on a chilly evening in the free state , it simply can’t be beat .
* awards : the 2006 william a . hipp scholarship recipient at the master brewers association of america’s ( mbaa ) concise course in brewing and malting science .
at free state , we don’t always strictly adhere to rules about beer styles , and we aren’t tied to marketing surveys . our mission is much simpler than that : provide good folks with great - tasting beer that is perfectly suited for almost any occasion .
after stints at maritime pacific brewing co . , liberty malt supply , pyramid breweries and pike brewing co . , geoff was experienced in seattle beer circles . but after seven years out west , he knew it was time to come back to the free state – and that’s just what he did . since signing on as a brewer in 2002 , geoff has continued to master his craft under the direction of our head brewer , steve bradt . he’s won some awards , * been invited to judge some pretty prestigious events * * and even took over as the big kahuna at our downtown brewery in 2009 . geoff was also a contributing writer to the oxford companion to beer .
yep – he’s come a pretty long way since that summer busing tables back in the day , and we’re proud as can be to call him a free stater .
nice head that dies fairly quick , beer is a deep dark amber in color . smells sweet of slight hops and breadiness . tastes nicely balanced between malt and hops with a bit of breadiness and slight sweetness . free state calls this one an amber ale . overall it ' s a nicely balanced easy drinker , a well done beer .
today , chuck can be found roaming the halls of the brewpub or the bottling facility , making sure his passion and knowledge come through in every pint served at the bar and bottle shipped to your local liquor store . if you see him , say hi . because after all , good beer and good conversation are what free state is all about .
if you talk to steve for even a few minutes , it becomes obvious how he’s managed to thrive . he’s clearly one sharp guy , especially when it comes to life science , history and all things mechanical . this , combined with his tireless study and refinement of the brewing process , makes him quite useful to have in your corner .
harvest festivals around the world have set the stage for beer fests . whether it ' s munich , denver , lawrence , manhattan or wichita , there is a full line - up of free wheeling , beer loving celebrations to satisfy your enthusiasm for a world of beer flavors .
geoff may have been born in iola , ks , but he was raised right here in lawrence , graduating from lawrence high in ’90 ( while holding down a summer job busing tables at the brewery ) and the university of kansas in ’95 . after getting his degree in art history ( a lot of good that did him ) geoff took off for seattle to pursue a career in brewing .
this ale is the first brewed in kansas since pioneering days . the name
ad astra comes from the kansas state motto - ad astra per aspera , latin words meaning
to the stars through difficulties
. we ' ve blended pale , caramel , and munich malts for a rich amber beer . the hops balance this malty sweetness , and we ' ve chosen northern brewer and fuggles hops for their flavor .
in english , explain why you ' re giving this rating . your review must discuss the beer ' s attributes ( look , smell , taste , feel ) and your overall impression in order to indicate that you have legitimately tried the beer . nonconstructive reviews may be removed without notice and action may be taken on your account .
i didn ' t care for it . it isn ' t sweet or bitter . . . . it just is . not really bad , but not something i ' d buy again either . i drank it cold straight from the bottle .
this brew is a clear and very dark color . the head is a creamy tan color and the aroma is a little malty , with toffee and cracker . . . the taste is drier than i was expecting , with some malt and roasted notes . a little caramel . the finish is , again , dry . just moderate
look : 2 | smell : 3 | taste : 3 . 5 | feel : 3 | overall : 3 . 5
pours a rich bloody - burgundy with a finger of khaki head . head foams up nicely , leaving a solid ring of foam around the top of the beer . lacing is strewn around the glass after each gulp . aroma brings about a barrage of lighthearted malts in the form of toasted and slightly bready notes . earthy hops sit right below . flavor profile opens up wonderfully , with notes of toasted and breaded malts , and earthy hops yet again . caramel and toffee notes peak out ever so slightly at the height of the flavor profile . mouth feel has a slight bit of grain to it , otherwise the carbonation adds some texture . it has a smooth , fairly crisp , medium - thin consistency . overall , a solid beer . it focuses on a select few notes , and delivers well . it ' s easy to gulp with its thinner feel , and tasty flavor .
l : small head that leaves a little bit of lacing . body is a dark copper color .
t & f : very malty and toffee - like up front . dark fruity finish . some bitterness in the aftertaste but doesn ' t linger very long . moderate carbonation and body .
look : 4 | smell : 3 . 25 | taste : 4 | feel : 4 | overall : 3 . 75
nice beer . i am no expert but it ' s very drinkable and is not a one and done type beer .
beautiful color . pairs great with food . not a strong alcohol taste . the price is even more attractive .
founded in 1996 , beeradvocate ( ba ) is your go - to resource for beer .
beeradvocate is a proud supporter of independent craft brewers . we encourage you to # seektheseal and drink independent beer .
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we ' re always out and about . come see what we ' re up to next .
our 14 - barrel brewhouse is made up of some of the best equipment available to the small brewer , but we still think the most important ingredients are the care and respect that our enthusiastic , knowledgeable staff has for each and every beer that we make . and now that we’re producing bottled beers in our expanded bottling facility , our attention to detail and dedication to great taste and quality can be experienced by people all over kansas , nebraska and missouri .
because without beer , things do not seem to go as well .
you can now check our beer list on the go when you visit our facebook page and click on the
use app
button . you ' ll be redirected to a live view of our menu , including abvs , ibus , brief descriptions , and pricing hosted by our friends at untappd .
because without beer , things do not seem to go as well .
want to check out our list right here on our website ? just click below .
this boldly hopped ale will take you to new heights of hoppiness . copious amounts of chinook , columbus , and cascade hops combine to evoke far off places with the scents and flavors of tropical fruit , pine needles , flowers and a hint of freshly mown grass . a choice malt bill lifts the hops , providing balance and keeping this beer on course .
by the time you’re done exploring our menu , you will be . click a section to get started .
from our lively spot in lawrence , kansas , we have been serving up great food and great beer to great people for over 25 years now . our home on historic massachusetts street is still where all of our newest brews are crafted , and where many of our most loyal customers enjoy their brews alongside a fresh batch of black bean quesadillas , crispy fish & chips or one of our featured daily specials .
our brewers are always busy thinking up fresh new brews . come see what we’re pouring today .
and while you may think that breaking even the stupidest , most controversial laws may be wrong , you can’t help but admire the passion . and that same passion , passed down two generations , is what drove chuck to change the laws in topeka , and open the first brewery in kansas in over a century .
meet steve . he’s our brewmaster , and he’s been hangin’ around since 1989 , when he took the role as our first full - time assistant brewer .
steve has been a judge at the great american beer festival and the world beer cup , and he holds the rank of national judge with the beer judge certification program . he’s also on the board of directors of the brewers association and serves on their events committee , which plans the great american beer festival , world beer cup and the craft brewer’s conference . steve is also the president of the kansas craft brewers guild .
there’s a rumor going around that he also sleeps from time to time , but we’re not sure we believe it .
* * judging : geoff has thrice been invited to judge at the great american beer festival , the nation’s largest and most prestigious beer festival and competion , and also at the world beer cup .
think i ' ve ever had . . . i even commented on it . locally sourced product . . .
we had heard good things and decided to give it a try . we were very disappoint
ely . i wouldn’t recommend the penne primavera or the gumbo . the gumbo tastes like hormel canned chili 🤭 . i wi
g my son’s 19th birthday in lawrence because my oldest son goes to ku and he’s too busy to come home . we were seated upstairs
in a semi private room with a table of 8 . it was loud , but everyone was just having a good time . the beer was great as always . i had a yakimaniac
e sour and ended with oatmeal stout . we had a burger , a rueben , a turkey sandwich , fish and chips and my son had the steak special , all were great as usual . the server heard us say “birthday boy” and brought my son a scoop of ice cream with a candle on it and the whole room sang him happy birthday , which embarrasse
d him a little and made him laugh a lot . i took a couple of spoonfuls of my son’s birthday ice cream and finished off dinner with an oatmeal stout flout . great night , a lot of fun , thanks fsbc . i promise to bring him back when he’s 21 !
excellent food , great price and friendly service . it is loud and can be hard to have a conversati
. ) they offer some tasty vegan options , including a black bean burger and hummus . i just wish that hot mulled win
i have not been here since before facebook existed , but it was great back in the mid 90 ' s . going to make back to the old stomping grounds and check out a ku basketball
service was slow , server was mia and my appetizer was cold as ice . my entire ' was pretty good . overall , mediocre experience
always great service & food here , even when they are very busy . never disappoint
s in the midwest . no matter where i travel in the world , i always compare all beers to my beloved copperhead
being from the pacific northwest where we have many many breweries i can honestly say that my cousins brewery is at the top of the list . the food was great and the beer was better . if you are anyw
my happy place ! great beer . great food . great service . great people .
and this is why we started putting our beer in cans ! enjoy a yaki on the lake today friends 🇺🇸 🍻 💥 🎥 - luke l .
in english , explain why you ' re giving this rating . your review must discuss the place ' s attributes and your overall impression in order to indicate that you have legitimately visited the place . nonconstructive reviews may be removed without notice and action may be taken on your account .
this faithful rendition of an american pale ale features a floral nose and a good deal of zesty citrus - like character in the finish . columbus , centennial , idaho 7 , and simcoe hops play together with the highest fidelity . light bodied with a golden orange hue .
this quick sour beer is light in body and big on flavor with a tart finish accentuated by guava fruit added to the conditioning vessel .
this is the strongest and arguably the most flavorful of the english style ales . a special treat made annually and set aside to age ( sometimes as long as ten years ) , this complex and warming beer is full of malt and hop character married in harmony .
this one has hops on the brain . . . … [ + ] expand description
this beer has hops on the brain to be sure , with additions throughout the process totaling close to three pounds per barrel . unfiltered , golden - hued , and extremely dry , this ipa features the lemony , citrusy character of centennial and citra hops grown in the hop capital of north america , yakima , wa .
our pilsner has a pleasant malt flavor , a medium body , and a finely balanced hop character . choice pilsner malt combines with prized german saphir hops .
extremely balanced , with a hint of hops . … [ + ] expand description
a full bodied ale , with a hint of hops to balance the sweetness that comes from three different malt varieties . the perfect way to take your taste buds to the stars .
light and refreshing golden beer brewed in a style all our own . goes down easy and finishes clean .
fine hop character , with a distinctive citrusy aroma and rich flavor . … [ + ] expand description
fine hop character , with a distinctive citrusy aroma and rich flavor . this pale ale has as almost as much bite as the slithering serpent who was the inspiration for its name .
in kansas , like other states in the great plains , the words “amber waves of grain” come alive . and as people who are no strangers to farms and farming ourselves , we thought it valuable to craft a beer that is made from the grain that kansas is known for , and positively perfect for quenching your thirst at the end of a long , hard day of work – or play .