RENDICONTI LINCEI MATEMATICA E APPLICAZIONI

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1 ATTI ACCADEMIA NAZIONALE LINCEI CLASSE SCIENZE FISICHE MATEMATICHE NATURALI RENDICONTI LINCEI MATEMATICA E APPLICAZIONI Alberto De Sole, Victor G. Kac On integral representations of -gamma and -beta functions Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Serie 9, Vol. 6 (25), n., p. 29. Accademia Nazionale dei Lincei < > L utilizzo e la stampa di uesto documento digitale è consentito liberamente per motivi di ricerca e studio. Non è consentito l utilizzo dello stesso per motivi commerciali. Tutte le copie di uesto documento devono riportare uesto avvertimento. Articolo digitalizzato nel uadro del programma bdim (Biblioteca Digitale Italiana di Matematica) SIMAI & UMI

2 Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Accademia Nazionale dei Lincei, 25.

3 Rend. Mat. Acc. Lincei s. 9, v. 6:-29 25) Funzioni speciali. ÐOn integral representations of -gamma and -beta functions. Nota *) di ALBERTO DE SOLE evictor G. KAC, presentata dalsocio C. De Concini. ABSTRACT. Ð We study -integralrepresentations of the -gamma and the -beta functions. As an application of these integral representations, we obtain a simple conceptual proof of a family of identities for Jacobi triple product, including Jacobi's identity, and of Ramanujan's formula for the bilateral hypergeometric series. KEY WORDS: -calculus; -gamma function; -beta function. RIASSUNTO. Ð Sulle rappresentazioni integrali delle funzioni -gamma e -beta. Studiamo la rappresentazione -integrale delle funzioni -gamma e -beta. Questo studio svela una -costante molto interessante. Come applicazione di ueste rappresentazioni integrali, otteniamo una semlice dimostrazione concettuale di una famiglia di identitaá per il prodotto triplo di Jacobi, che include l'identitaá di Jacobi, e della formula di Ramanujan per le serie ipergeometriche bilaterali.. INTRODUCTION There is no generalrigorous definition of a «-analogue». An intuitive definition of a -analogue of a mathematical object A is a family of objects A,< <, such that the limit of A,as tends to, is A. Under certain additionalreuirements the -analogue may be uniue, but sometimes it is usefulto consider several-analogues of the same object. The -calculus i.e. the -analogue of the usual calculus) begins with the definition of the -analogue d f x) of the differentialof a function, df x). The former is much simpler than the latter: d f x) f x) f x) : Having said this, we immediately get the -analogue of the derivative of f x), called its - derivative: : D f x) : d f x) d x f x) f x) )x Notice that the -derivative satisfies the following -analogue of Leibniz rule :2 D f x)g x)) g x)d f x) f x)d g x) : Of course, the first uestion is what is the -derivative of x a, where a 2 C. This is very easy: D x a [a]x a ; : *) Pervenuta in forma definitiva all'accademia il 27 luglio 24.

4 2 A. DE SOLE - V.G. KAC where [a] is the -analogue of a: [a] a : Here we discover the remarkable property of a -analogue A of an object A - sometimes A makes sense even when A does not: [] : The next uestion is what is the -analogue x a) n of the function x a)n for a nonnegative integer n. The naturalreuirements are :3 D x a) n [n] x a) n ; x a) : It is a nice exercise to check that the uniue) answer is n 2 N) :4 x a) n Yn j x j a) : The product.4) plays in combinatorics the most fundamental role. Again, remarkably, for x, a x this formula makes sense for n : :5 x) Y j j x) : Under our assumptions on, the infinite product.5) is convergent. Now we may give a definition of x) a for any number a, which is consistent with.4): x) a x) :6 a x) : We shall need the following generalizations of.3), where n 2 Z and a; b; t 2 C, 8 D ax b) n a[n] ax b) n ; :7 >< >: D a bx) n b[n] a bx) n ; D bx) t b[t] bx) t : The proof of these identities is left as a simple exercise for the reader. There are two important -analogues of the exponential function: :8 E x X n n )=2 xn )x) ; [n]! n :9 e x X n x n [n]! )x) ;

5 ON INTEGRAL REPRESENTATIONS OF -GAMMA AND -BETA FUNCTIONS 3 where [n]! [][2] [n]. Note that the series in.8) resp..9)) converges for jxj < resp. jxj < []. The second eualities in.8) and.9) are due to Euler. The simplest proof of them is obtained by using the -analogue of Taylor's formula see e.g. [6]). The function e x is uniuely defined by the reuirements D e x ex ; e : However, due to the product expansions in.8) and.9), e x ) E x not e x ), which explains why one needs both -analogues of the exponential function. The -derivative of E x is D E x E x : In the present paper we shall discuss -analogues of Euler's gamma and beta functions G t) and B t; s). Their definitions are reminded in Section 2, euations 2.) and 2.2) for the unlikely reader who is not familiar with these functions). The -gamma function was introduced by Thomae [4] and later by Jackson [4] as the infinite product : G t) )t ) t ; t > : Though the literature on the -gamma function and its applications is rather extensive see [-3] and references there), the authors usually avoided the use of its -integral representation. In fact, each time when a -integralrepresentation was discussed, it was, as a rule, not uite right. The first correct integral representation of G t) that we know of is in references [, 2]: : G t) Z Š x t E x d x ; where the -integral introduced by Thomae [4] and Jackson [5]) is defined by :2 Z a f x)d x ) X j a j f a j ) : Notice that the series on the right-hand side is guaranteed to be convergent as soon as the function f is such that, for some C > ; a >, j f x)j < Cx a in a right neighborhood of x. The definition.2) has an obvious interpretation in terms of a Riemann sum. Moreover it is easy to check that the -integraldefined in.2) is the -antiderivative. Namely for any function f x) continuous at x we have :3 Z a D f x)d x f a) f ) ; D Z x f t)d t f x) :

6 4 A. DE SOLE - V.G. KAC In the following we will denote, for arbitrary numbers a; b Z b a f x)d x Z b Z a f x)d x f x)d x : As immediate conseuence -Leibniz rule.2) and of euation.3), we get the - analogue of the rule of integration by parts :4 Z b a Z g x)d f x)d x f x)g x) b b f x)d g x)d x : a The -beta function was more fortunate than the -gamma function as far as the integralrepresentations are concerned. Already in the above mentioned papers by Thomae and Jackson it was shown that the -beta function, defined by the -analogue of the usualformula 2.5), B t; s) G s)g t) :5 ; G s t) has the following -integralrepresentation, which is a -analogue of Euler's formula 2.2): :6 B t; s) Z x t x) s d x ; t; s > : Jackson [5] made an attempt to give a -analogue of a less traditional Euler's integral representation of the beta function: :7 B t; s) Z x t a t s dx ; x) which is obtained from 2.2) by the substitution x! = x). However, his definition is not uite right, since it is not uite eualto B t; s), as will be explained in Remark 3.5. A correct -analogue of.7) is the famous Ramanujan's formula for the bilateral hypergeometric series, see [, pp ]. In the present paper we give another -integralrepresentation of G t), based on the -exponentialfunction e x,anda-integralrepresentation of B t; s) whichisaanalogue of.7). Both representations are based on the following remarkable function, cf. [2]: K x; t) xt x t :8 x) t : x This function is a -constant in x, i.e. K x; t) K x; t) ; and for t anintegeritisindependentofxand is eualto t t )=2. However, for t 2 ; )

7 ON INTEGRAL REPRESENTATIONS OF -GAMMA AND -BETA FUNCTIONS 5 this function does depend on x, sincefortheset one has lim K x; t) x t x t :! Our integralrepresentations are as follows: :9 G t) K A; t) =A ) Z x t e x d x ; :2 B t; s) K A; t) Z=A x t x) t s d x ; where the improper integral, following [5] and [3], is defined by Z=A f x)d x ) X n A f n :2 : A n2z Since K A; t) depends on A, we conclude that the integrals in both formulas do depend on A. In his formulas Jackson used the factor t t )=2 in place of K A; t), which is correct only for an integer t. For the proof of these integralrepresentations, provided in Section 3, we willneed the following reciprocity relations, which follow immediately from the definition of -integral: :22 Z A f x)d x Z=A =A x 2 f d x ; x Z=A f x)d x Z A x 2 f d x : x These are change of variables rules under the substitution x!, and similar rules are x easily derived for the substitution x! ax b. However there is no generalchange of variables formula for the -integral. This is the main handicap of -calculus. The reader may find more on -calculus and its applications in the books [, 3, 6]. In Sections 4 and 5 we will apply euation.2) to find an integral representation of the -beta function which is manifestly symmetric under the exchange of t and s, and to find a -analogue of translation invariance of certain improper integrals. Finally, in Section 6 we will show that euation.9) is euivalent to a family of triple product identities, a limiting case of which is the Jacobi triple product identity: :23 ) x) =x) X n ) n n n )=2 x n ; and that euation.2) is euivalent to Ramanujan's summation formula, see [, pp. 5-55]: :24 X n a) n b) n x n ) b=a) ax) =ax) b) =a) x) b=ax) :

8 6 A. DE SOLE - V.G. KAC Also, the symmetric integral representation of the -beta function is euivalent to the following identity: :25 X n a) n =a) n b) n c) n a) =a) b) c) ) bc=) b=a) ac=) : Jacobi's triple product identity and Ramanujan's summation formula are among the most famous and important identities in all of Mathematics. In particular, each of these identities turned out to be the tip of the iceberg of an important representation theory: The Jacobi triple product identity respectively a specialization of the Ramanujan's summation formula) is the «denominator identity» of the simplest affine Kac-Moody algebra, sl 2)b resp. of the simplest affine superalgebra, sl 2; )b), [7, 9]. At the same time, these identities arise in uantum field theory, as an essential part of the bosonfermion correspondence and its super analogue [8]. One way or another most of the results of the paper have appeared in the literature, however our exposition seems to be more systematic. 2. DEFINITION OF -GAMMA AND -BETA FUNCTIONS The Euler's gamma and beta functions are defined as the following definite integrals s; t > ): 2: G t) Z x t e x dx ; 2:2 2:3 Z B t; s) x t x) s dx ; Z x t t s dx : x) From the expression 2.2) it is clear that B t; s) is symmetric in t and s. Recall some of the main properties of the gamma and beta functions: 2:4 2:5 G t ) tg t) ; G ) ; B t; s) G t)g s)=g t s) : In this paper we are interested in the -analogue of the gamma and beta functions. They are defined in the following way. DEFINITION 2.. a) For t >, the -gamma function is defined to be

9 ON INTEGRAL REPRESENTATIONS OF -GAMMA AND -BETA FUNCTIONS 7 2:6 G t) b) For s; t >, the -beta function is 2:7 B t; s) Z Z Š x t E x d x : x t x) s d x : G t) and B t; s) are the «correct» -analogues of the gamma and beta functions, since they reduce to G t) and B t; s) respectively in the limit!, and they satisfy properties analogues to 2.4) and 2.5). This is stated in the following 2:8 THEOREM 2.2. a) G t) can be euivalently expressed as In particular one has G t) )t ) t : G t ) [t]g t) ; 8t > ; G ) : b) The -gamma and -beta functions are related to each other by the following two euations G t) B t; ) 2:9 ) t ; 2: B t; s) G t)g s) G t s) : PROOF. We reproduce here the proof of Kac and Cheung [6, pp ], because similar arguments will be used to prove the results in the next section. If we put s in the definition of the -beta function, use.8) and the change of variable x )y,we get B t; ) Z x t x) d x Z x t E x d x = ) Z ) t y t E y d y ) t G t) ; which proves 2.9). It follows from -integration by parts.4) and euation.7) that B t; s) satisfies the following recurrence relations t; s > ):

10 8 A. DE SOLE - V.G. KAC 2: B t ; s) Z x t D x) s [s] x B t; s ) The above euations 2.) imply [s] Z D x t ) x) s d x [t] [s] B t; s ) ; Z Z x t x) s d x x t s x) x) s d x B t; s) s B t ; s) : B t; s ) [s] [s t] B t; s) : Since clearly B t; ), we get, for t > and any positive integer n, [t] 2:2 B t; n) [n ]...[] [t n ]...[t] )n ) t ) ) t n Taking the limit for n!in this expression we get B t; ) ) ) t : )n ) t ) n This together with 2.9) proves 2.8). Now we prove 2.). By comparing 2.2) and 2.8) we have that 2.) is true for any positive integer value of s. To conclude that 2.) holds for non integer values of s we will use the following simple argument. If we substitute a s and b t in 2.) we can write the left-hand side as : ) X n b n n ) a n ) ; and the right-hand side as ) ) ab) a) b) :

11 ON INTEGRAL REPRESENTATIONS OF -GAMMA AND -BETA FUNCTIONS 9 Both these expressions can be viewed as formalpower series in with coefficients rational functions in a and b. Since we already know that they coincide, for any given b, for infinitely many values of a of the form a n, with positive integer n), it follows that they must be eualfor every value of a and b. This concludes the proof of the theorem. p 3. AN EQUIVALENT DEFINITION OF -GAMMA AND -BETA FUNCTIONS In the previous section the definition of G t) was obtained from the integral expression 2.) of the Euler's gamma function, simply by replacing the integral with a Jackson integraland the exponentialfunction e x with its -analogue E x. It is naturalto ask what happens if we use the other -exponentialfunction. In other words, we want to study the following function A > ): 3: =A ) Z g A) t) x t e x d x : Similarly, the function B t; s) was obtained by taking the -analogue of the integral expression 2.2) of the Euler's beta function. We now want to study the -analogue of the integralexpression 2.3). We thus define 3:2 b A) t; s) Z =A x t x) t s d x : In this section we will show how the functions g A) t)andb A) t; s) are related to the - gamma and -beta function respectively. We will adapt in this situation the same arguments used for the proof of Theorem 2.2. Recall the main steps used there. put s in the definition of B t; s) to derive euation 2.9), 2. use -integration by parts to derive recurrence relations for B t; s), 3. notice that B t; s) is a formalpower series in with coefficients rationalfunctions in a s and b t. Step. By taking the limit s! in the definition of b A) t; s), using the infinite product expansion of e x and making the change of variables x )y, we get b A) t; ) Z =A x t x) d x Z=A x t e x d x We therefore proved ) t =A ) Z y t e y d y ) t g A) t) :

12 2 A. DE SOLE - V.G. KAC 3:3 g A) t) ) t b A) t; ) : Step 2. We now want to find recursive relations for g A) t) and b A) t; s). By integration by parts.4) we get g A) t ) t [t]g A) t) : Here we used the fact that x t e x tends to zero as x! and x! the second fact follows from the identity e x =E x ). Since obviously g A) ), we conclude that for every positive integer n and any value of A > ), 3:4 n n )=2 g A) n) [n ]! G n) : Let us now consider the function b A) t; s). In order to perform integration by parts we will need the following LEMMA 3.. For arbitrary numbers a; b we have D x a x) b [a] x a x) b [b] [a]) x a x) b : PROOF. The lemma follows immediately from the definition.) of -derivative, and the definition.6) of x) a. p It then follows from integration by parts.4) and Lemma 3. that, for t; s > 3:5 b A) t ; s) [t s] t Z=A x) t D x) t s d x Z=A t [t s] x) t s t [t] [t s] b A) t; s) : D x t d x For t we have 3:6 b A) ; s) Z =A x) s d x [s] : Formulas 3.5) and 3.6) imply s > ; n 2 Z ) 3:7 n n )=2 b A) )s n; s) ) ) n ) s n B n; s) :

13 ON INTEGRAL REPRESENTATIONS OF -GAMMA AND -BETA FUNCTIONS 2 Similarly we have by integration by parts.4) and Lemma 3. 3:8 b A) t; s ) [t s] s We now need to compute b A) =A Z 3:9 b A) t; ) Z=A Z=A s [t s] [s] [t s] b A) t; s) : x) s D x t s x) t s x t s x) t s d x D x s d x t; ). By definition of b A) t; s) and Lemma 3. we have x t x) t d x Z=A [t] D x t x) t d x : When using the fundamentaltheorem of -calculus to compute the right-hand side, we have to be careful, since the limit for x! of the function F x) x) t does not exist. On the other hand, by definition of -derivative and Jackson integral, we have Z=A D F x)d x lim F N! A N lim F N ; N! A where the limits on the right-hand side are taken over the seuence of integer numbers N. We then have from 3.9) b A) t; ) lim [t] N! AN ) t 3: A N )t : Let us denote by K A; t) the limit in parenthesis in the right-hand side of 3.). We can compute the limit for N!in the expression of K A; t), simply using the definition.6) of x) a : K A; t) A t N =A =A =A lim N! Nt t N =A t =A t =A A t t A A N lim A N! A t A N t A At t A) t : A From 3.8) and 3.) we conclude that for any t > and positive integer n K A; t)b A) )n t; n) ) ) t 3: ) n t B t; n) : xt

14 22 A. DE SOLE - V.G. KAC In the following lemma we enumerate some interesting properties of the function K x; t) x xt t x) t : x LEMMA 3.2. a) In the limits! and! we have lim K x; t) ; 8 x; t 2 R! lim K x; t)! xt x t ; 8 t 2 ; ); x 2 R : In particular K x; t) is not constant in x. b) Viewed as a function of t, K x; t) satisfies the following recurrence relation: K x; t ) t K x; t) : Since obviously K x; ) K x; ), we have in particular that for any positive integer n K x; n) n n )=2 : c) As function of x, K x; t) is a «-constant», namely D K x; t) ; 8 t; x 2 R : In other words K n x; t) K x; t) for every integer n. PROOF. The limit for! ofk x; t) is obviously. In the limit! we have, for any a >, x) a x) x) a x)! x) : We therefore have, for t 2 ; ): lim K x; t) x t. For part b) we need to use the! x following obvious identity y) a a y) y) a : It follows from the definition of K x; t) that K x; t ) t x xt x x) t x t x t x) K x; t) t K x; t) : For part c) it suffices to prove that K x; t) K x; t). By definition K x; t) x x)t t x) t : x

15 ON INTEGRAL REPRESENTATIONS OF -GAMMA AND -BETA FUNCTIONS 23 We can replace in the right-hand side t x = x) t t = x) x ; x) t t x x) t : x The claim follows from the following trivial identity t x t x) : x) t x This concludes the proof of the lemma. p REMARK 3.3. The function K x; t) is an interesting example of a function which is not constant in x and with -derivative identically zero. Step 3. It follows from 3.7), 3.) and Lemma 3.2 that the functions K A; t)b A) t; s) and B t; s) coincide for any A > as soon as either t or s is a positive integer. We want to prove that they actually coincide for every t; s >. THEOREM 3.4. For every A; t; s > one has: 3:2 K A; t)g A) t) G t) ; 3:3 K A; t)b A) t; s) B t; s) : REMARK 3.5. This result corrects and generalizes a similar statement of Jackson [5]. There 3.3) is proved in the specialcase in which s t is a positive integer, But, due to a computationalmistake, the factor K A; t) is missing. PROOF OF THEOREM 3.4. Euation 3.2) is an immediate corollary of 2.9), 3.3) and 3.). As in the proof of Theorem 2.2, in order to prove 3.3) it suffices to show that K A; t)b A) t; s) can be written as formalpower series in with coefficients rationalfunctions in a s and b t. After performing a change of variable y Ax, we get 3:4 K A; t)b A) t; s) A t Z = A) t A y t t s y A d y :

16 24 A. DE SOLE - V.G. KAC Fix A >. After letting b t, we can rewrite the factor in front of the integralas A A) A b ; A A b and this is manifestly a formal power series in with coefficients rationalfunctions in b. Wethenonlyneedtostudytheintegral term in 3.4), which we decompose as 3:5 Z y t t s y A d y Z = y t t s y A After letting a s and b t the first term in 3.5) can be written as ) X n b n ab A ; n n A andthisisaformalpowerseriesinwith coefficients rationalfunctions in a and b. We are left to consider the second term in 3.5). By relation.22) we can rewrite it as 3:6 Z x s x t s Ax t s d x : Recalling the definition of K x; t), we have the identity x t s Ax t s Ax Ax) t s d y: A t s K Ax; t s) : The main observation is that, even though K Ax; t s) is not constant in x, by Lemma 3.2 K A n ; t s) K A; t s); 8 n 2 Z, therefore inside the Jackson integralit can be treated as a constant. Using this fact, we can rewrite 3.6) as A t s Z 3:7 K A; t s) Ax xs Ax) t s d x : After letting a s and b t we can finally rewrite the first factor in 3.7) as A) ab A A ab 3:8 A A ; and the integralterm in 3.7) as 3:9 ) X a n An 2 ) : n An 2 ab Clearly both expression 3.8) and 3.9) are formal power series in with coefficients rationalfunctions in a and b. This concludes the proof of the theorem. p

17 ON INTEGRAL REPRESENTATIONS OF -GAMMA AND -BETA FUNCTIONS 25 REMARK 3.6. The proof of Theorem 3.4 is more complicated than one may have expected. This is due to the fact that we do not have a formula for the change of variables x! x. Notice that, instead, we took advantage of the fact that one can take out of the sign of -integralan arbitrary -constant, not just a constant as for ordinary integrals. 4. APPLICATION : AN INTEGRAL EXPRESSION OF THE -BETA FUNCTION MANIFESTLY SYMMETRIC IN t AND s Theorem 2:2 implies that B t; s) is a symmetric function in t and s. This is not obvious from its integralexpression 2.7). We now want to use Theorem 3.4 to find an integral expression for B t; s) which is manifestly symmetric under the exchange of t and s. By Theorem 3.4 we have that, for any A > 4: B t; s) K A; t) Z=A x t x) t s d x : By definition of K x; t) we get, after simple algebraic manipulations 4:2 x)t s xt K x ; t t x t t x) s : Since by Lemma 3.2 we have K x ; t K A; t) ; 8 x n A ; n 2 Z ; when we substitute 4.2) back into 4.) we get, after a change of variable y t x, 4:3 B t; s) Z =a y t d y ; 8 a > : y) s y To conclude, we just notice that this integral expression of B t; s) is manifestly symmetric in t and s, since performing the change of variable x namely applying the reciprocity y relation.22)) gives the same integral with a replaced by =a and s replaced by t. 5. APPLICATION 2: TRANSLATION INVARIANCE OF A CERTAIN TYPE OF IMPROPER INTEGRALS One of the main failures of the Jackson integral is that there is no analogue of the translation invariance identity

18 26 A. DE SOLE - V.G. KAC Z a f x)dx Z a c c f x c)dx ; obviously true for «classical» integrals. By using Theorem 3.4 we are able to write a - analogue of translation invariance for improper integrals of a special class of function, namely of the form x a = x) b. More precisely we want to prove the following 5: COROLLARY 5.. For a > and b > a we have Z=A x a x) b d x Z = x a b K A; a) a x x b d x : REMARK 5.2. In the «classical» limit, the right-hand side is obtained from the left-hand side by translating x! x. PROOF. From the definition of B t; s) we have 5:2 Z B t; s) x s x) t d x Z = s x s x Z = t d x x t t x x t s d x : The first identity was obtained by applying.22) and the second by a change of variable y x=. From Theorem 3.4 we also have 5:3 B t; s) K A; t) Z=A x t x) t s d x : Euation 5.) is obtained by comparing 5.2) and 5.3), after letting a t ; b t s and using the fact that K A; a ) a K A; a). p REMARK 5.3. As we have remarked above, the -integralis not translation invariant. However, as pointed out in [] see also [3]), translation invariance can sometimes be restored by working with -commuting variables. T.H. Koornwinder also pointed out to us that formula 3.3) is essentially euivalent to exercise 6.7 i) from [3].

19 ON INTEGRAL REPRESENTATIONS OF -GAMMA AND -BETA FUNCTIONS APPLICATION 3: IDENTITIES If we rewrite euations 3.2), 3.3) and 4.3) using the definition of improper integrals, we get some interesting identities involving -bilateral series. After using the infinite product expansion.9) of the -exponentialfunction e x, the expression 2.8) of the -gamma function, the definition.2) of the improper Jackson integral and simple algebraic manipulations, we can rewrite euation 3.2) as 6: ) t =A) A=t ) A) t ) X n tn =A) n : If we let x t =A in euation 6.) we get 6:2 ) x) =x) A) Ax) X n x) n A n =A) n : This is a -parameter family of identities for the well known Jacobi triple product ) x) =x) ; parametrized by A. Notice that lim A! An =A) n n n )=2 : This implies that, in the limit A!, euation 6.2) reduces to the famous Jacobi triple product identity.23). Let us consider now euation 3.3). After using the definition.2) of improper - integral, the expression.5) for the -beta function and simple algebraic manipulations, we can rewrite it as 6:3 X n =A) n t s =A) n tn ) t s ) t =A) A=t ) t s =A) A) t ) s ) : Notice that, after letting a =A; b t s =A; x t, euation 6.3) is euivalent to the famous Ramanujan's identity.24). In other words, the proof of Theorem 3.4 in Section 3 can be viewed as a new conceptualproof of Ramanujan's identity. Finally we can rewrite euation 4.3) as 6:4 X n =a) n a) n s =a) n t a) n =a) a) s =a) t a) ) t s ) s ) t ) : After letting a =a; b s =a; c t a, euation 6.4) reduces to.25). REMARK 6.. It is also interesting to see what are the identities corresponding to the originalintegralexpression of the -gamma and -beta functions. For this, expand the integralexpression 2.6) of the -gamma function using the definition of -integral, use

20 28 A. DE SOLE - V.G. KAC the infinite product representation.8) of the -exponentialfunction and compare the result with 2.8). The resulting identity is: t ) X n nt ) n which is, after replacing x t = ), the same as Euler's identity.9). Similarly, expand the integralexpression 2.7) of the -beta function and use 2.) to get ; X n nt s ) n ) n s t ) t ) : If we then make the change of variables a s ; x t, we get the well known Heine's product formula for a -hypergeometric series: ax) x) X a) n ) n x n : n ACKNOWLEDGEMENTS We thank T.H. Koornwinder for interesting comments on the originalversion of this paper. One of the authors wishes to thank A. Varchenko for sending corrections to the book [6]. REFERENCES [] G.E. ANDREWS - R. ASKEY - R. ROY, Special functions. Encyclopedia of Mathematics and its Applications, vol., Cambridge University Press, Cambridge 999, 664 pp. [2] H. EXTON, -hypergeometric functions and applications. Ellis Horwood Series in Mathematics and its Applications, Ellis Horwood Ltd., Chichester 983, 347 pp. [3] G. GASPER -M.RAHMAN, Basic hypergeometric series. Encyclopedia of Mathematics and its Applications, vol. 35, second edition, Cambridge University Press, Cambridge 99, 456 pp. [4] F.H. JACKSON, A generalization ofthe functions G n) and x n. Proc. Roy. Soc. London, 74, 94, [5] F.H. JACKSON, On -definite integrals. Quart. J. Pure and Applied Math., 4, 9, [6] V.G. KAC - P. CHEUNG, Quantum Calculus. Universitext, Springer-Verlag, New York 22, 2 pp. [7] V.G. KAC, Infinite dimensional Lie algebras. Third edition, Cambridge University Press, Cambridge 99, 4 pp. [8] V.G. KAC, Vertex algebras for beginners. University Lecture Series,, second edition, American MathematicalSociety, Providence, RI, 998, 2 pp. [9] V.G. KAC - M. WAKIMOTO, Integrable highest weight modules over affine superalgebras and number theory. In: J.-L. BRYLINSKI et al. eds.), Lie theory and geometry. Progr. Math., 23, BirkhaÈuser, Boston 994, [] A. KEMPF - S. MAJID, Algebraic -integration and Fourier theory on uantum and braided spaces. Journalof MathematicalPhysics, 35, 994, [] T.H. KOORNWINDER, -special functions, a tutorial. Preprint math. CA/94326, 994. [2] T.H. KOORNWINDER, Compact uantum groups and -special functions. In: V. BALDONI - M.A.

21 ON INTEGRAL REPRESENTATIONS OF -GAMMA AND -BETA FUNCTIONS 29 PICARDELLO eds.), Representations oflie groups and uantum groups. Pitman Research Notes in Mathematics Series, 3, Longman Scientific & Technical, 994, [3] T.H. KOORNWINDER, Special functions and -commuting variables. In: M.E.H. ISMAIL - D.R. MASSON - M. RAHMAN eds.), Special functions, -series and related topics Toronto, ON, 995). Fields Inst. Commun., vol. 4, Amer. Math. Soc., Providence 997, [4] J. THOMAE, Beitrage zur Theorie der durch die Heinesche Reihe. J. reine angew. Math., 7, 869, Pervenuta il24 giugno 24, in forma definitiva il 27 luglio 24. A. De Sole: Department of Mathematics Harvard University Oxford St - CAMBRIDGE, MA 238 USA) desole@math.harvard.edu V.G. Kac: Department of Mathematics, MIT 77 Massachusetts Avenue - CAMBRIDGE, MA 239 USA) kac@math.mit.edu

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