arXiv:1805.12548v1 [math.CA] 19 May 2018 6 ,1,1,1,2] n rcinlcluu 1 6 17]. 16, [1, monot Γ( calculus 7], function, fractional 5, Gamma and [2, The 23], inequalities 14, 22], dig 11, 15, 10, of 13, development 8, 12, [6, of [9, theory properties the to in func respect participated zeta have Riemann’s who func as Authors, Digamma such t functions applications. etc. in the special function used of many mainly t range with is in wide connected It application a as with [0,1]. used functions domain be special the to for function models digamma distributions-probability of property natural A Introduction 1 diga of study further hypergeometr the in Clausen’s useful for potentially digam theorems are of here An summation presented values series. some the hypergeometric present computed of concept have som to the established we using we also integers, Here, and positive authors. function, many by digamma interest for of impact high a o,Elrsitga ftescn id sdfie by defined is kind) second the of sets, Euler’s (or, the on function rnae as yegoercseries hypergeometric Gauss Truncated Abstract: ue-aceoicntn;Gmafunction. Gamma constant; Euler-Mascheroni n15,Kr eesrs aeanvldfiiino am functio gamma of definition novel a gave Weierstrass Karl 1856, In rnae as yegoercsre n t application its and series hypergeometric Gauss Truncated 2010 e od n phrases and words Key ahmtc ujc Classification Subject -al:miqureshi E-mails: ntels eae,teter fdgmafnto a endeve been has function digamma of theory the decades, last the In aut fEgneigadTcnlg,ai ilaIslamia Millia Technology,Jamia and Engineering of Faculty ..Qrsi am ae n .Shadab M. and Jabee Saima Qureshi, M.I. eateto ple cecsadHumanities, and Sciences Applied of Department ACnrlUiest) e Delhi-110025,India. New University), Central (A Γ( Γ( 1 iam Pi ucin eeaie yegoercseries; hypergeometric Generalized function; (Psi) Digamma : z z ) = ) z ,wsitoue yLoadElra eeaiaino the of generalization a as Euler Leonard by introduced was ), n [email protected]. and = lim = R [email protected],[email protected], [email protected], ∗ z iam function digamma Z n orsodn author Corresponding →∞ x ( exp 0 falra ubr,and numbers, real all of ∞ x ( exp Z 18,3B5 18,33C05. 11J86, 33B15, 11J81, : γz 0 n )  − n Y 1 ∞ =1 t ) − t h z − n t + 1 1  dt, n t z n z −  1 dt. exp ℜ C  ( z − falcmlxnmes It numbers. complex all of ) n z > i m function. mma tep a enmade been has attempt 0 n , nct 1–1,series [18–21], onicity cfnto.Results function. ic mafnto with function amma ∗ neetn results interesting e inadClausen’s and tion eter fbeta of theory he in r directly are tions afnto for function ma eter of theory he oe with loped (1.2) (1.1) in 1 Truncated Gauss hypergeometric series 2 where γ = 0.577215664901532860606512090082402431042 ... is called Euler-Mascheroni 1 constant, and Γ(z) is an entire function of z, and

1 1 1 γ = lim 1+ + + ..... + − ln n . n→∞  2 3 n 

The function ψ(z) is the of the or digamma func- tion or Psi-function, given by

d Γ′(z) ψ(z)= {ln Γ(z)} = , (1.3) dz Γ(z)

z ln Γ(z)= ψ(ζ)dζ. (1.4) Z1

The widely-used Pochhammer symbol (λ)ν (λ, ν ∈ C) is defined by

1 (ν = 0; λ ∈ C \{0}) Γ(λ + ν)  (λ)ν := = (1.5) Γ(λ)  λ (λ + 1) ... (λ + n − 1) (ν = n ∈ N; λ ∈ C) ,  it is being understood conventionally that (0)0 = 1 and assumed tacitly that the Γ quo- tient exists.

The generalized pFq, is defined by

∞ m (ap); [(ap)]m z pFq z = (1.6)  (bq);  [(b )] m! mX=0 q m • p and q are positive integers or zero,

• z is a complex variable,

• (ap) designates the set {a1, a2, ..., ap},

• the numerator parameters a1, ..., ap ∈ C and the denominator parameters C Z− b1, ..., bq ∈ \ 0 ,

r

• [(ar)]k = (ai)k. By convention, a product over the empty set is 1. Yi=1

Thus, if a numerator parameter is a negative integer or zero, the pFq series terminates, then we are led to a generalized hypergeometric polynomial. Truncated Gauss hypergeometric series 3

2 Truncated Gauss series and its application in digamma function

In 1931-32, W. N. Bailey (see, [3, p. 40, last eq.] and [4, p. 34, last eq.] ) derived a formula for truncated Gauss hypergeometric series in terms of Clausen’s hypergeometric series as follows

a, b; a, b; F 1 = Sum of first (n+1)-terms of series F 1 2 1 f 2 1 f  ; n  ;  n (a) (b) = k k (f)kk! Xk=0 Γ(a + n + 1)Γ(b + n + 1) a, b, f + n; = F 1 , (2.1) Γ(n + 1)Γ(a + b + n + 1) 3 2  f, a + b + n + 1;  where f ≥ a + b. It can be written as

n a, b, f + n; Γ(n + 1)Γ(a + b + n + 1) (a)k(b)k 3F2 1 = , (2.2)  f, a + b + n + 1;  Γ(a + n + 1)Γ(b + n + 1) (f)kk! Xk=0 where f ≥ a + b. Putting f = a + b + 1 in above equation (2.2), we get

a, b; Γ(a + n + 1)Γ(b + n + 1) a, b, a + b + n + 1; F 1 = F 1 2 1 a b 3 2 a b , a b n  + + 1; n Γ(n + 1)Γ(a + b + n + 1)  + +1 + + + 1;  Γ(a + n + 1)Γ(b + n + 1) a, b; = F 1 Γ(n + 1)Γ(a + b + n + 1) 2 1  a + b + 1;  (2.3)

Using Gauss summation theorem, we get

a, b; (a + 1) (b + 1) F 1 = n n (2.4) 2 1 a b  + + 1; n (a + b + 1)n n! In (2.2), put a =1, b =1, f = 2 and n = m − 1, we get

m−1 m 1, 1, m + 1; m +1 (1)k m +1 1 3F2 1 = = , m =1, 2, 3,..., (2.5)  2, m + 2;   m  (2)k  m  k Xk=0 Xk=1 it is the simplest way to calculate the Clausen’s 3F2-series for positive integers.

Now we calculate the values of Clausen’s series 3F2 by applying the result (2.5). Truncated Gauss hypergeometric series 4

In (2.5), put m = 1, we get 1 1, 1, 2; 2 1 F 1 = =2. (2.6) 3 2  2, 3;  1 k Xk=1 In (2.5), put m = 2, we get 2 1, 1, 3; 3 1 9 F 1 = = . (2.7) 3 2  2, 4;  2 k 4 Xk=1 In (2.5), put m = 3, we get 3 1, 1, 4; 4 1 22 F 1 = = . (2.8) 3 2  2, 5;  3 k 9 Xk=1

Similarly for higher values of m (m =4, 5,..., 51) in (2.5), the values of 3F2, are calculated using wolfram mathematica, and arranged in Table1, Table2 and Table3. Now, we establish an interesting formula for the computation of digamma function using Clausen’s hypergeometric function given by (2.5).

Let us recall the Weierstrass definition of gamma function (1.2)

∞ 1 z z = z exp (γz) 1+ exp − Γ(z) n n nY=1 h   i Applying ln on both sides with respect to the base ’e’ ∞ z z ln 1 − ln Γ(z) = ln z + γz ln e + ln 1+ exp − (2.9) n n Xn=1 n   o or, ∞ z z − ln Γ(z) = ln z + γz + ln 1+ − (2.10) n n Xn=1 n   o Differentiating with respect to ’z’, we get ′ ∞ Γ(z) 1 1 1 ψ(z)= = − − γ − − Γ(z) z n + z n Xn=1 ′ ∞ Γ(z) 1 z ψ(z)= = − − γ + (2.11) Γ(z) z (n + 1)(n + z + 1) Xn=0 Now writing R.H.S. of (2.11) in hypergeometric notation using (1.6), we get

′ 1, 1, 1+ z; Γ(z) 1 z ψ(z)= = − − γ + 3F2  1 , (2.12) Γ(z) z 1+ z 2, 2+ z;   where z =06 , −1, −2, −3,... , and ψ(z) denotes the digamma function.

For z ∈ N in (2.12), and using the values of 3F2 arranged in Tables (1),(2),(3), we can find ψ(1), ψ(2), ψ(3),...,ψ(52) (see Table (4)). Truncated Gauss hypergeometric series 5

3 Some new summation theorems

Table 1: ( Clausen’s summation theorem ) S.No. Clasuen’s series S.No. Clasuen’s series 1, 1, 3; 1, 1, 13; 9 1118273 1 3F2  1 = 4 11 3F2  1 = 332640 2, 4; 2, 14;     1, 1, 4; 1, 1, 14; 22 1145993 2 3F2  1 = 9 12 3F2  1 = 334620 2, 5; 2, 15;     1, 1, 5; 1, 1, 15; 125 1171733 3 3F2  1 = 48 13 3F2  1 = 336336 2, 6; 2, 16;     1, 1, 6; 1, 1, 16; 137 2391514 4 3F2  1 = 50 14 3F2  1 = 675675 2, 7; 2, 17;     1, 1, 7; 1, 1, 17; 343 41421503 5 3F2  1 = 120 15 3F2  1 = 11531520 2, 8; 2, 18;     1, 1, 8; 1, 1, 18; 726 42142223 6 3F2  1 = 245 16 3F2  1 = 11571560 2, 9; 2, 19;     1, 1, 9; 1, 1, 19; 6849 271211719 7 3F2  1 = 2240 17 3F2  1 = 73513440 2, 10; 2, 20;     1, 1, 10; 1, 1, 20; 7129 275295799 8 3F2  1 = 2268 18 3F2  1 = 73717644 2, 11; 2, 21;     1, 1, 11; 1, 1, 21; 81191 11167027 9 3F2  1 = 25200 19 3F2  1 = 2956096 2, 12; 2, 22;     1, 1, 12; 1, 1, 22; 83711 18858053 10 3F2  1 = 25410 20 3F2  1 = 4938024 2, 13; 2, 23;     Truncated Gauss hypergeometric series 6

Table 2: ( Clausen’s summation theorem ) S.No. Clasuen’s series S.No. Clasuen’s series 1, 1, 23; 1, 1, 33; 439143531 586061125622639 21 3F2  1 = 113809696 31 3F2  1 = 140027687654400 2, 24; 2, 34;     1, 1, 24; 1, 1, 34; 1332950097 53676090078349 22 3F2  1 = 342075734 32 3F2  1 = 12741489961200 2, 25; 2, 35;     1, 1, 25; 1, 1, 35; 33695573875 54062195834749 23 3F2  1 = 8561966208 33 3F2  1 = 12752521554240 2, 26; 2, 36;     1, 1, 26; 1, 1, 36; 34052522467 54437269998109 24 3F2  1 = 8580495000 34 3F2  1 = 12762940281000 2, 27; 2, 37;     1, 1, 27; 1, 1, 37; 309561680403 2027671241084233 25 3F2  1 = 77338861600 35 3F2  1 = 472593445833600 2, 28; 2, 38;     1, 1, 28; 1, 1, 38; 312536252003 2040798836801833 26 3F2  1 = 77445096300 36 3F2  1 = 472938908878800 2, 29; 2, 39;     1, 1, 29; 1, 1, 39; 9146733078187 2053580969474233 27 3F2  1 = 2248776129600 37 3F2  1 = 473266655870400 2, 30; 2, 40;     1, 1, 30; 1, 1, 40; 9227046511387 2066035355155033 28 3F2  1 = 2251453244040 38 3F2  1 = 473578015512420 2, 31; 2, 41;     1, 1, 31; 1, 1, 41; 288445167734557 85205313628946333 29 3F2  1 = 69872686884000 39 3F2  1 = 19428841662048000 2, 32; 2, 42;     1, 1, 32; 1, 1, 42; 581548514594714 85691034670497533 30 3F2  1 = 139890941865675 40 3F2  1 = 19440406448751600 2, 33; 2, 43;     Truncated Gauss hypergeometric series 7

Table 3: ( Clausen’s summation theorem ) S. No. Clasuen’s series 1, 1, 43; 75614351220200831 41 3F2  1 = 17069625174513600 2, 44;   1, 1, 44; 5853599356775405587 42 3F2  1 = 1315072372819818600 2, 45;   1, 1, 45; 5884182435213075787 43 3F2  1 = 1315751996785100160 2, 46;   1, 1, 46; 5914085889685464427 44 3F2  1 = 1316402071882326000 2, 47;   1, 1, 47; 279336945645849479669 45 3F2  1 = 61900150757844484800 2, 48;   1, 1, 48; 280682601097106968469 46 3F2  1 = 61928185246412349150 2, 49;   1, 1, 49; 13818010880930033053031 47 3F2  1 = 3035798698036894732800 2, 50;   1, 1, 50; 13881256687139135026631 48 3F2  1 = 3037063614161076772272 2, 51;   1, 1, 51; 13943237577224054960759 49 3F2  1 = 3038278925731369320000 2, 52;   1, 1, 52; 14004003155738682347159 50 3F2  1 = 3039447494548958308200 2, 53;   Truncated Gauss hypergeometric series 8

4 Application in digamma function

Table 4: Digamma function for positive integers ′ ′ Γ (z) Γ (z) z ψ(z)= Γ(z) z ψ(z)= Γ(z) 34395742267 1 −γ 27 −γ + 8923714800 312536252003 2 −γ +1 28 −γ + 80313433200 3 315404588903 3 −γ + 2 29 −γ + 80313433200 11 9227046511387 4 −γ + 6 30 −γ + 2329089562800 25 9304682830147 5 −γ + 12 31 −γ + 2329089562800 137 290774257297357 6 −γ + 60 32 −γ + 72201776446800 49 586061125622639 7 −γ + 20 33 −γ + 144403552893600 363 53676090078349 8 −γ + 140 34 −γ + 13127595717600 761 54062195834749 9 −γ + 280 35 −γ + 13127595717600 7129 54437269998109 10 −γ + 2520 36 −γ + 13127595717600 7381 54801925434709 11 −γ + 2520 37 −γ + 13127595717600 83711 2040798836801833 12 −γ + 27720 38 −γ + 485721041551200 86021 2053580969474233 13 −γ + 27720 39 −γ + 485721041551200

1145993 2066035355155033 14 −γ + 360360 40 −γ + 485721041551200

1171733 2078178381193813 15 −γ + 360360 41 −γ + 485721041551200

1195757 85691034670497533 16 −γ + 360360 42 −γ + 19914562703599200

2436559 12309312989335019 17 −γ + 720720 43 −γ + 2844937529085600

42142223 532145396070491417 18 −γ + 12252240 44 −γ + 122332313750680800

14274301 5884182435213075787 19 −γ + 4084080 45 −γ + 1345655451257488800

275295799 5914085889685464427 20 −γ + 77597520 46 −γ + 1345655451257488800

55835135 5943339269060627227 21 −γ + 15519504 47 −γ + 1345655451257488800

18858053 280682601097106968469 22 −γ + 5173168 48 −γ + 63245806209101973600

19093197 282000222059796592919 23 −γ + 5173168 49 −γ + 63245806209101973600

444316699 13881256687139135026631 24 −γ + 118982864 50 −γ + 3099044504245996706400

1347822955 13943237577224054960759 25 −γ + 356948592 51 −γ + 3099044504245996706400

34052522467 14004003155738682347159 26 −γ + 8923714800 52 −γ + 3099044504245996706400 Truncated Gauss hypergeometric series 9

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