Some Extensions of the Prabhu-Srivastava Theorem Involving the (p, q)-gamma Function

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1 Filomat 31: ), Published by Faculty of Scieces ad Mathematics, Uiversity of Niš, Serbia Available at: Some Extesios of the Prabhu-Srivastava Theorem Ivolvig the p, )-Gamma Fuctio Ji-Cai Liu a, Jichu Liu b a College of Mathematics ad Iformatio Sciece, Wezhou Uiversity, Wezhou , PR Chia b School of Mathematics Zhuhai), Su Yat-Se Uiversity, Zhuhai , PR Chia Abstract. I this paper, we obtai some it formulas for derivatives of p, )-gamma fuctio ad p, )- digamma fuctio at their poles. These it formulas exted the Prabhu-Srivastava theorem ivolvig gamma fuctio ad digamma fuctio. 1. Itroductio It is well-ow that for all complex umbers x 0, 1, 2,, the gamma fuctio ad digamma fuctio [1, pp. 255] are defied by! x Γx) = xx + 1) x + ) ad ψx) = Γ x) Γx), respectively. A. Prabhu ad H.M. Srivastava [8] have cosidered the its of ratios betwee two gamma fuctios ad digamma fuctios at their poles x = 0, 1, 2,, ad obtaieome ice formulas: Theorem 1.1. [8, Theorem 1 ad 2]) For o-egative iteger ad positive itegers ad m, we have ad Γx) Γmx) = m 1) m) m)! )!, 1) ψx) ψmx) = m Mathematics Subject Classificatio. Primary 33D05; Secodary 33B15 Keywords. p, )-Gamma fuctio; -Gamma fuctio; Faà di Bruo formula; Prabhu-Srivastava theorem; Derivative Received: 06 April 2016; Accepted: 20 July 2016 Commuicated by Hari M. Srivastava Research supported by Natural Sciece Foudatio of Fujia Provice No. 2015J05002), ad Fujia Provicial Foudatio of Educatio ad Research for Youg Teachers No. JA15531). addresses: jc2051@163.com Ji-Cai Liu), liujc1982@126.com Jichu Liu)

2 J.-C. Liu, J. Liu / Filomat 31: ), Applyig 1) ad Gauss-Legedre multiplicatio formula, they also obtaied a iterestig product idetity for the gamma fuctio: 1 Γ + j ) = 1) 1) π) 2 1)! )!, j=1 for o-egative iteger ad positive iteger 2. I 2013, F. Qi [9] cosidered the its of ratios betwee two derivatives of gamma fuctio ad digamma fuctio at their poles. Theorem 1.2. [9, Theorem 1.2]) For o-egative itegers s, ad positive itegers, m, we have ad ) x) m s+1 mx) = 1) m) m)! )!, 2) ψ s) x) ) m s+1 ψ s) mx) =. 3) Remar. Theorem 1.2 is cotaied i the Prabhu-Srivastava theorem Theorem 1.1) by obvious use of the L Hôpital s rule for its. For a o-egative iteger p, the p-gamma fuctio is defied by Γ p x) = p!p x xx + 1) x + p), 4) which was first itroduced by Euler. Similarly, the p-digamma fuctio is give by ψ p x) = Γ px) Γ p x). Note that p Γ p x) = Γx) ad p ψ p x) = ψx), ad both Γ p x) ad ψ p x) are aalytic o the complex plae except for x = 0, 1, 2,, p. Recetly, L. Yi ad L.-G. Huag [5] provided alterative proofs of 1) ad 3) by establishig the followig results: Theorem 1.3. [5, Theorem 2.3 ad 2.6]) Let, p, s be o-egative itegers ad m, be positive itegers such that m, p. The Γ p x) Γ p mx) = m ) ) p p p)m ) /, 5) m ad ψ s) p x) ) m s+1 ψ s) p mx) =. 6) Lettig p i 5) ad 6) ad otig that ) ) p p p pm ) / = m)! m )!, we are led to 1) ad 3). They also posed the followig cojecture:

3 J.-C. Liu, J. Liu / Filomat 31: ), Cojecture 1.4. [5, Cojecture 2.9]) Let s, ad p be o-egative itegers ad m, be positive itegers such that m, p. The p x) ) m s+1 ) p p p mx) = p) m ) /. 7) ) m It is ot hard to see that 7) reduces to 2) whe p. Remar. Theorem 1.3 ad Cojecture 1.4 ca be cosidered as the p-extesios of the Prabhu-Srivastava theorem Theorem 1.1). F. H. Jacso defied the followig -gamma fuctios [4, I.35), pp.353]: ad Γ x) = ; ) x ; ) 1 ) 1 x for 0 < < 1, Γ x) = 1 ; 1 ) x ; 1 ) 1) 1 x x 2) for > 1, 8) where a; ) 0 = 1 ad a; ) = 1 =0 1 a ). This fuctio have may aalogues of the classical facts about the gamma fuctio [2, 7]. Similarly, the -digamma fuctio is give by ψ x) = Γ x) Γ x). It is well-ow that 1 Γ x) = Γx) ad 1 ψ x) = ψx), ad both Γ x) ad ψ x) have the poles at x = 0, 1, 2,. V.B. Krasii, H.M. Srivastava ad S.S. Dragomir[3] cosidered the followig p, )-gamma fuctio ad p, )-digamma fuctio: x 1 2 ) [p] x [p]! Γ p, x) = for > 1, 9) [x] [x + 1] [x + p] ad ψ p, x) = Γ p,x)/γ p, x), where p is a o-egative iteger ad [x] = 1 x )/1 1 ). They have also obtaieome complete mootoicity properties of the p, )-gamma fuctio. Both Γ p, x) ad ψ p, x) have the poles at x = 0, 1,, p. Note that 9) reduces to 4) whe 1, ad reduces to 8) whe p. I this paper, we shall establish some extesios of the Prabhu-Srivastava theorem Theorem 1.1) ivolvig the p, )-gamma fuctio. We will see that all of Theorem 1.1, 1.2, 1.3 ad Cojecture 1.4 are special cases of these theorems. 2. Statemets of the Results We ca rewrite 9) as Γ p, x) = 1 1 ) 1 ; 1 ) p x ; 1 ) p+1 1 p 1 1 It is clear that the defiitio 10) is euivalet to ) x x 1 2 ) for > 1. 10) Γ p, x) = 1 ); ) p[p] x x 1 2 ) x ; ) p+1 for < 1, 11) where [p] = 1 p )/1 ). I what follows we will use the defiitio 11) for Γ p, x).

4 J.-C. Liu, J. Liu / Filomat 31: ), Theorem 2.1. Let s, ad p be o-egative itegers ad, m be positive itegers such that, m p. The ψ s) ) p,x) m s+1 ψ s) p,mx) =. 12) Lettig 1 i 12), we obtai 6). Theorem 2.2. Let s, ad p be o-egative itegers ad, m be positive itegers such that, m p. The ) p,x) m s+1 [ [ ] p,mx) = ) m ) p p [p] /, 13) ] m where [p] = 1 p )/1 ) ad the -biomial coefficiet is give by [ ] a ; ) a =. b ; ) b ; ) a b Lettig 1 i 13), we obtai 7), ao we cofirm Cojecture 1.4. Theorem 2.3. Let a, b be positive itegers a, ad p be o-egative itegers such that p. The p, x) a x) = b a a b)+1)2 p, b ) 1 a +1 1 bp 1 b 1 ap ) / a. 14) b 3. Proof of the Results I order to prove the results, we eeome importat lemmas. Lemma 3.1. Faà di Bruo) If ad f are fuctios with a sufficiet umber of derivatives, the f x)) = dxs This is the famous Faà di Bruo formula [6]. s! r 1! r 2! r s! r 1+r 2 + +r s ) f x)) f 1) ) r1 x) f s) ) rs x). 15) 1! s! Lemma 3.2. Let F ad log F be fuctios with a sufficiet umber of derivatives. For ay positive iteger s, there exist some coefficiets ar 1, r 2,, r s ) idepedet of x such that F s) x) = Fx) ar 1, r 2,, r s ) f 1) x) ) r 1 f s) x) ) r s, 16) where f x) = log Fx). Proof. Lettig x) = e x ad f x) = log Fx) i 15), we immediately get F s) x) = Fx) This completes the proof. s! r 1! r s! f 1) ) r1 x) f s) ) rs x). 1! s!

5 Proof of Theorem 2.1. By 11), we have ) p, x) = ds 1 dx log[p] s 1 + ds 1 3 dx s 1 2 x log Lettig x) = log x ad f x) = 1 i+x i 15) gives dx s log1 x+i ) = log ) s J.-C. Liu, J. Liu / Filomat 31: ), p i=0 s! r 1! r 2! r s! dx s log1 x+i ). 17) R 1)! 1!) r 1 2!) r 2 s!) r s x+i)r 1 x+i ) R, 18) where R = r 1 + r r s. Sice 1 r r s r s = s, R has the maximum value R = s whe r 1 = s ad r 2 = = r s = 0, ao we ca write 18) i the form dx s log1 x+i ) = log ) s s x+i)r C s R) 1 x+i ), 19) R R=0 where C s R) is idepedet of x ad C s s) 0. Note that 19) has the pole at x = i. Combiig 17) ad 19), we have for s 1 ad, m p, Notig that p, x) p, mx) = x+)s 1 x+) ) s 1 mx+) ) s mx+)s. 1 mx+) = m 1 x+), 20) we obtai p, x) p, mx) = m ) s for s 1, which is euivalet to 12). Proof of Theorem 2.2. We first prove the case s = 0. Γ p, x) Γ p, mx) = m+2 2 ) +2 2 ) [p] m ) Applyig 20) ad otig that i ; ) i = 1) i i+1 2 ) ; )i, we obtai Γ p, x) Γ p, mx) = m [ ] ) m ) p [p] m ; ) m ; ) p m ; ) ; ) p z 1 mx+) 1 x+). /. 21) m Let f p, x) = log Γ p, x). By 16), we have p,x) = Γ p, x) ar 1, r 2,, r s ) f 1) p, x) ) r 1 f p,x) ) s) r s. 22)

6 Notig that f d) p, x) = ψ d 1) p, x) ad the usig 12), we get f d) p, x) f d) p, mx) = m It follows from 22) ad 23) that J.-C. Liu, J. Liu / Filomat 31: ), ) d for d 1. 23) p,x) p,mx) = Γ p, x) ) m s Γ p, mx). 24) The proof of 13) the directly follows from 21) ad 24). Proof of Theorem 2.3. We first prove the case s = 0. Γ p, ax) Γ p, bx) = a b)+2 2 ) Usig 20) ad otig that we obtai ) 1 a +1 1 bp 1 b 1 ap a ; a ) p b ; b ) b ; b [ ] ) p = a b)+1 b ; b ) p a ; a ) a ; a 2 ) p ) p Γ p, ax) Γ p, bx) = b a a b)+1)2 a / ) 1 a +1 1 bp 1 b 1 ap ) a ; a ) p b ; b ) b ; b ) p b ; b ) p a ; a ) a ; a ) p 1 bx+) 1 ax+)., b ) / a I order to prove 14), by 16) ad 25), it suffices to prove that log Γp, ax) ) s). 25) b log Γp, bx) ) s) = 1 for s 1. Replacig by a i 19) yields dx s log1 ax+i) ) = a log ) s s ax+i)r C s R) 1 ax+i) ). R R=0 Similarly to the proof of Theorem 2.1, we have log Γp, ax) ) s) log Γp, bx) ) s) = a b ) s ax+)s 1 ax+) ) 1 bx+) ) s = 1 by 20)). s bx+)s This completes the proof. Acowledgmets. The authors would lie to tha the referee for valuable commets which improved the presetatio of the paper.

7 J.-C. Liu, J. Liu / Filomat 31: ), Refereces [1] M. Abramowitz ad I. A. Stegu, Hadboo of Mathematical Fuctios with Formulas, Graphs, ad Mathematical Tables, Natioal Bureau of Stadards, Applied Mathematics Series 55, 9th pritig, Washigto, [2] R. Asey, The -gamma ad -beta fuctios, Appl. Aal ), [3] V.B. Krasii, H.M. Srivastava ad S.S. Dragomir, Some complete mootoicity properties for the p, )-gamma fuctio, Appl. Math. Comput ), [4] G. Gasper ad M. Rahma, Basic hypergeometric series, Cambridge Uiversity Press, [5] L. Yi ad L.-G. Huag, Limit formulas related to the p-gamma ad p-polygamma fuctios at their sigularities, Filomat ), [6] W. P. Johso, The curious history of Faà di Bruo s formula, Amer. Math. Mothly ), [7] D.S. Moa, The -gamma fuctio for > 1, Aeuatioes Math ), [8] A. Prabhu ad H.M. Srivastava, Some it formulas for the gamma ad psi or digamma) fuctios at their sigularities, Itegral Trasforms Spec. Fuct ), [9] F. Qi, Limit formulas for ratios betwee derivatives of the gamma ad digamma fuctios at their sigularities, Filomat ),