Improved q-exponential and q-trigonometric functions

Impro v ed q -exp onen tial and q -trigonometric function s Jan L. Cie ´ sli ´ nski ∗ Uniw ersytet w Bia lymstoku, Wydzia l Fizyki ul. Lipow a 41, 1 5-424 Bia lystok, Poland Abstract W e prop ose a new definition of t he q -exp onential function. Our q -exp onential function maps the imaginary axis into the un it circle and the resulting q -trigonometric functions are b oun ded and satisfy the Pythagorean iden tit y . MSC 2010: 05A30; 33B10; 39A13 Key wor ds an d ph r ases: quan tum calculus, q - calculus, q -exp onential func- tion, q -t r ig onometric functions, Ca yley transform, ∗ e-mail: j anek @ alp ha.uwb .edu.p l 1 1 In tro duction The quan tum calculus ( q -calculus) is an old, classical branc h of mathemat- ics, whic h can b e traced ba c k to Euler and Gauss [11, 21] with imp ortant con tributio ns of Jac kson a century ago [18, 19]. In recen t y ears there a re man y new deve lopmen ts and applications of the q -calculus in mathematical ph ysics, especially concerning sp ecial functions [1, 8, 12, 14] and quan tum mec hanics [4, 5, 25, 1 3, 10, 23, 29]. Many pap ers w ere dev oted to v arious approac hes to q -deformations of elemen tary functions, including exp onential and trigonometric functions [2 , 3, 7, 15, 22, 2 4, 26, 27, 28]. In this pap er we prop ose new definitions of the q -exponential function and q -trigonometric functions. These r esults are motiv a ted by recen t dev el- opmen ts in the time scales calculus, where new exponential, h yp erb olic and trigonometric function ha ve b een defined [9 ]. The concept of time scales uni- fies difference and differen tia l calculus [16]. The q -calculus can b e considered as a calculus on a sp ecial time scale (see, e.g., [6]). The functions presen ted in this pap er hav e b etter qualitativ e prop erties than standar d q -exponen tial and q -trigonometric functions. In order to dis- cuss and compare these properties we begin with a short summary of the classical results, usually follo wing the textb o ok [20 ]. In the standard approac h to the q -calculus t w o exp onen tia l f unction are used: e z q = ∞ X n =0 z n [ n ]! , E z q = ∞ X n =1 z n [ ˜ n ]! , (1) where q is p ositiv e, z is comple x, and [ n ]! = [1][2] . . . [ n ] , [ k ] = 1 + q + q 2 + . . . + q k − 1 , [ ˜ n ]! = [ ˜ 1][ ˜ 2] . . . [ ˜ n ] , [ ˜ k ] = 1 + 1 q + 1 q 2 + . . . + 1 q k − 1 . (2) Hence we immediately get E z q = e z 1 /q . Another, more po pula r , form of E z q is obtained using the iden tit y [ ˜ n ]! = q (1 − n ) n 2 [ n ]! . (3) Both exp o nen tial functions can b e represen ted by infinite pro ducts, e z q = ∞ Y k =0 (1 − (1 − q ) q k z ) − 1 , E z q = ∞ Y k =0 (1 + (1 − q ) q k z ) . (4) 2 F rom this form w e easily see that e z q E − z q = 1 . Moreo v er, D q e z q = e z q , D q E z q = E q z q , (5) where D q ( q - deriv ative or Jac kson’s deriv ativ e) is defined b y D q f ( z ) := f ( q z ) − f ( q ) q z − z . (6) The existence o f tw o represen tations of q -exp onential functions (infinite series and infinite pro duc t) is related to w ell known formulae for the usual exp o nen tial function ( q = 1), e z = ∞ X k =0 z k k ! = lim m →∞  1 + z m  m . (7) Tw o expo nen tial functions of the quantum calculus generate tw o pairs of the q - trigonometric functions. Using notation of [20] w e ha v e: sin q x = e ix q − e − ix q 2 i , Sin q x = E ix q − E − ix q 2 i , cos q x = e ix q + e − ix q 2 , Cos q x = E ix q + E − ix q 2 . (8) T aking in to accoun t prop ertie s of q -exp onen tia l functions (see ab ov e) w e easily deriv e prop erties of standard q -trigonometric functions: cos q x Cos q x + sin q x Sin q x = 1 , sin q x Cos q x = cos q x Sin q x , (9) D q sin q x = cos q x , D q cos q x = − sin q x , D q Sin q x = Cos q ( q x ) , D q Cos q x = − Sin q ( q x ) . (10) Note that the corresp onding t angen ts coincide: T an q x = tan q x . 3 2 Impro v e d q -exp one ntial func t ion New q -exp onential function E z q is defined as E z q := e z 2 q E z 2 q = ∞ Y k =0 1 + q k (1 − q ) z 2 1 − q k (1 − q ) z 2 , (11) where e z q , E z q are standard q -exp onential functions. This definition is moti- v ated b y the class ical Cay ley transformation z → ca y ( z , a ) := 1 + az 1 − az , (12) see, e.g., [9, 17]. Indeed, E q z q = 1 − (1 − q ) z 2 1 + (1 − q ) z 2 E z q = ca y  − z 2 , 1 − q  E z q . (13) Theorem 1. The q -exp onential function E z q is analytic in the disc | z | < R q and E z q = ∞ X n =0 z n { n } ! , (14) for | z | < R q , wher e R q =                2 1 − q for 0 < q < 1 , 2 q q − 1 for q > 1 , ∞ for q = 1 , (15) { n } := 1 + q + . . . + q n − 1 1 2 (1 + q n − 1 ) = [ n ] 1 2 (1 + q n − 1 ) = 2(1 − q n ) (1 − q )(1 + q n − 1 ) , (16) and, final l y, { n } ! = { 1 }{ 2 } . . . { n } . 4 Pr o of: In the d isc | z | < 1 b o th series (1 ) are absolutely con vergen t for any q ∈ R + . Multiplying th em we get e z 2 q E z 2 q = ∞ X k =0 ∞ X j =0 q j ( j − 1) 2  z 2  k + j [ k ]![ j ]! = ∞ X n =0  z 2  n [ n ]!   n X j =0 q j ( j − 1) 2 [ n ]! [ j ]! [ n − j ]!   . (17) Using Gauss’s binomial formula (see, e.g. , [20], formula (5.5)) n − 1 Y j =0 ( z + aq j ) = n X j =0 q j ( j − 1) 2 [ n ]! [ j ]! [ n − j ]! a j z n − j , (18) w e hav e, as a p articular case, n X j =0 q j ( j − 1) 2 [ n ]! [ j ]! [ n − j ]! = (1 + 1)(1 + q ) . . . (1 + q n − 1 ) . (19) Substituting (19) in to (17) w e get the form u la (14) with { n } defined by (16). In order to obtain the radius of conv ergence, w e compu te lim n →∞     z n +1 { n + 1 } !         { n } ! z n     = lim n →∞     z { n + 1 }     =    (1 − q ) | z | 2 for q < 1 ( q − 1) | z | 2 q for q > 1 (20) Then, using d ’Alem b ert’s test, w e get (for q 6 = 1) the radius of conv ergence (1 5). Note that R 1 /q = R q . In the case q = 1 all q -exponential f unctions coincide with e z , hence R 1 = ∞ . ✷ Theorem 2. The q -exp on e n tial function E z q has the fol lowing p r op erties: E − z q =  E z q  − 1 , | E ix q | = 1 , (21) E z q = E z 1 /q , D q E z q = h E z q i , (22) wher e z ∈ C , x ∈ R and we use the notation h f ( z ) i := f ( z ) + f ( q z ) 2 . Pr o of: The fi rst equation of (21) is a straigh tforward consequence of the definition (11). T hen, E z q = E ¯ z q . Hence, |E ix q | 2 = E ix q E ix q = E − ix q E ix q = 1 . (23) 5 The symb ol { n } dep ends on q . In this pro of it is con venien t to use m ore precise notation { n } ≡ { n } q , { n } ! ≡ { n } q !. The equation E z q = E z 1 /q follo ws immediately from the ob vious iden tit y { n } q ! = { n } 1 /q ! . (24) Finally , D q E z q = E q z q − E z q q z − z = E z q ( q − 1) z  1 − (1 − q ) z 2 1 + (1 − q ) z 2 − 1  = E q z q 1 + (1 − q ) z 2 , (25) hE z q i = 1 2  E q z q + E z q  = 1 2  1 − (1 − q ) z 2 1 + (1 − q ) z 2 + 1  E z q = E q z q 1 + (1 − q ) z 2 , (26) whic h implies the second equatio n of (22). ✷ The prop erties (21) are iden tical with analog ical prop erties of the ex- p onen tial function e z . W e po int out that neither e z q nor E z q satisfies (21). Instead, w e ha ve E − z q e z q = 1 . 3 Impro v e d q -trigonometric func t ions New q -sine and q -cosine functions are defined in a na tural w a y: S in q x = E ix q − E − ix q 2 i , C os q x = E ix q + E − ix q 2 . (27) Theorem 3. q -T rigonometric functions define d by ( 2 7) sa tisfy: C os 2 q x + S in 2 q x = 1 , D q S in q x = hC os q x i , D q C os q x = −hS in q x i , (28) Pr o of: Prop erties (28) follo w directly from (21), (22) (note that E ix q E − ix q = 1). ✷ Corollary 4. q -T rigonometric functions C os q x , S in q x ar e r e al for x ∈ R . Mor e over, for any x ∈ R , we have − 1 6 C os q x 6 1 , − 1 6 S in q x 6 1 . (29) 6 Theorem 5. New q -trigonometric functions c an b e expr esse d by standar d q -trigonom etric functions as fol lows: C os q 2 x = cos q x Cos q x − sin q x Sin q x = 1 − tan 2 q x 1 + tan 2 q x , S in q 2 x = sin q x Cos q x + cos q x Sin q x = 2 tan q x 1 + tan 2 q x . (30) Pr o of: First, we compute cos q x Cos q x − sin q x Sin q x = e ix q E ix q + e − ix q E − ix q 2 = C os q 2 x , sin q x Cos q x + cos q x Sin q x = e ix q E ix q − e − ix q E − ix q 2 = S in q 2 x . (31) Then, using (9), we get C os q 2 x = cos q x Cos q x − sin q x Sin q x cos q x Cos q x + sin q x Sin q x = 1 − tan q x T an q x 1 − tan q x T an q x , S in q 2 x = sin q x Cos q x + cos q x Sin q x cos q x Cos q x + sin q x Sin q x = tan q x + T an q x 1 − tan q x T an q x . (32) T aking in to acc oun t T an q x = tan q x we complete the pro of. ✷ 4 Conclus ions Motiv ated b y the class ical Cay ley transformation and recen t resu lts in the time scales calculus (see [9]), w e in tro duced a new definition of the q -exp o- nen tial function. Main adv an tages of the new q -exp o nen tial function consist in b etter qualitative prop erties (i.e., its prop ertie s are more similar to prop- erties of e z ). In particular, it maps the unitary axis in to the unit circle, compare (21 ) , whic h implies ex cellen t prop erties of new trigonometric func- tions, including fo rm ulae (28) and boundedness (29). Esp ecially inte resting is t he Pythagorean iden tity: C os 2 q x + S in 2 q x = 1. According to o ur b est know ledge, other q -defomations of trigonometric func- tions do not satisfy this prop erty . T he same concerns ev en the pap er [1 5], full of surprising iden tities. 7 Our exp onen tial function is closely related to b ot h p opular q -exponential functions (1). There fore, pro ofs a nd calculations concerning E z q can b e usually done with the help of kno wn results. W e plan to expres s in terms of the new exp o nen tial function classical results containing q - exp onen tial f unctions, a nd w e hop e to obtain some impro vem en ts. References [1] G.E.Andrews, R.Askey , R.Roy: S p e cial fun ct ions , Cambridge Univ. P ress 199 9. [2] N.M.A takishiyev: “On a one-par ameter family of q -exp onential functions” , J. Phys. A: Math. Gen. 29 (199 6) L22 3-L227 . 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