aa r X i v : . [ m a t h . G M ] J a n ON A NEW CLASS OF SERIES IDENTITIES
ARJUN K. RATHIE
Abstract.
The aim of this paper is to provide a new class of seriesidentities in the form of four general results. The results are establishedwith the help of generalizatons of the classical Kummer’s summationtheorem obtained earlier by Rakha and Rathie. Results obtained earlierby Srivastava, Bailey and Rathie et al. follow special cases of our mainfindings.
Primary : 33B20,33C20, Secondary : 33B15, 33C05
Keywords:
Generalized hypergeometric function, Kummer’s summa-tion theorem, product formulas, generalization, Double series Introduction and Results Required
We start with the following two very interesting results involving productof generalized hypergeometric series due to Bailey[1] viz. F (cid:20) − ρ ; x (cid:21) × F (cid:20) − ρ ; − x (cid:21) = F (cid:20) − ρ, ρ, ρ + ; − x (cid:21) (1.1)and F (cid:20) − ρ ; x (cid:21) × F (cid:20) − − ρ ; − x (cid:21) = F (cid:20) − , ρ + , − ρ ; − x (cid:21) + 2(1 − ρ ) xρ (2 − ρ ) F (cid:20) − , ρ + 1 , − ρ ; − x (cid:21) (1.2)Bailey[1] established these results with the help of the following classicalKummer’s summation theorem[2] viz. F (cid:20) a, b a − b ; − (cid:21) = Γ (cid:0) a (cid:1) Γ (1 + a − b )Γ (1 + a ) Γ (cid:0) a − b (cid:1) (1.3)Very recently, Rathie et al.[6] have obtained explicit expressions of(i) F (cid:20) − ρ ; x (cid:21) × F (cid:20) − ρ + i ; − x (cid:21) (ii) F (cid:20) − ρ ; x (cid:21) × F (cid:20) − ρ − i ; − x (cid:21) (iii) F (cid:20) − ρ ; x (cid:21) × F (cid:20) − − ρ + i ; − x (cid:21) (iv) F (cid:20) − ρ ; x (cid:21) × F (cid:20) − − ρ − i ; − x (cid:21) in the most general form for any i ∈ Z and provided the natural general-izations of the results (1.1) and (1.2).The aim of this paper is to obtain explicit expressions of(a) ∞ X m =0 ∞ X n =0 ( − n ∆ m + n x m + n ( ρ ) m ( ρ + i ) n m ! n !(b) ∞ X m =0 ∞ X n =0 ( − n ∆ m + n x m + n ( ρ ) m ( ρ − i ) n m ! n !(c) ∞ X m =0 ∞ X n =0 ( − n ∆ m + n x m + n ( ρ ) m (2 − ρ + i ) n m ! n !(d) ∞ X m =0 ∞ X n =0 ( − n ∆ m + n x m + n ( ρ ) m (2 − ρ − i ) n m ! n !in the most general form for any i ∈ Z . Here { ∆ m } is a sequence ofarbitrary complex numbers.The results are derived with the help of the following generalizations ofKummer’s summation theorem obtained earlier by Rakha and Rathie[4] for i ∈ Z viz. F (cid:20) a b a − b + i ; − (cid:21) = 2 − a Γ (cid:0) (cid:1) Γ( b − i )Γ(1 + a − b + i )Γ( b )Γ (cid:0) a − b + i + (cid:1) Γ (cid:0) a − b + i + 1 (cid:1) (1.4) × i X r =0 (cid:18) ir (cid:19) ( − r Γ (cid:0) a − b + i + r + (cid:1) Γ (cid:0) a − i + r + (cid:1) and F (cid:20) a b a − b − i ; − (cid:21) = 2 − a Γ (cid:0) )Γ(1 + a − b + i (cid:1) Γ (cid:0) a − b + i + (cid:1) Γ (cid:0) a − b + i + 1 (cid:1) (1.5) × i X r =0 (cid:18) ir (cid:19) Γ (cid:0) a − b − i + r + (cid:1) Γ (cid:0) a − i + r + (cid:1) Results obtained earlier by Srivastava[8], Bailey[1] and Rathie et al.[6] followspecial cases of our main findings.
N A NEW CLASS OF SERIES IDENTITIES 3 Main Results
The results to be established in this paper are given in the followingtheorem.
Theorem 2.1.
Let { ∆ n } be a bounded sequence of complex numbers. Thenfor i ∈ Z , the following general results hold true: ∞ X m =0 ∞ X n =0 ( − n ∆ m + n x m + n ( ρ ) m ( ρ + i ) n m ! n != ∞ X m =0 ∆ m x m ( ρ ) m m ! 2 m Γ (cid:0) (cid:1) Γ( ρ + i )Γ(1 − ρ − m − i )Γ(1 − ρ − m ) Γ (cid:0) ρ + i + m − (cid:1) Γ (cid:0) ρ + i + m (cid:1) × i X r =0 ( − r (cid:18) ir (cid:19) Γ (cid:0) ρ + m + i + r − (cid:1) Γ (cid:0) r − i − m + (cid:1) ! (2.1) ∞ X m =0 ∞ X n =0 ( − n ∆ m + n x m + n ( ρ ) m ( ρ − i ) n m ! n != ∞ X m =0 ( − n ∆ m x m ( ρ m m ! 2 m Γ (cid:0) (cid:1) Γ( ρ − i )Γ (cid:0) ρ − i + m − (cid:1) Γ (cid:0) ρ − i + m (cid:1) × i X r =0 (cid:18) ir (cid:19) Γ (cid:0) ρ + m − i + r − (cid:1) Γ (cid:0) r − i − m + (cid:1) ! (2.2) ∞ X m =0 ∞ X n =0 ( − n ∆ m + n x m + n ( ρ ) m (2 − ρ + i ) n m ! n != ∞ X m =0 ∆ m x m ( ρ ) m m ! 2 ρ − m Γ (cid:0) (cid:1) Γ(2 − ρ + i )Γ( − m − i )Γ( − m ) Γ (cid:0) m − ρ + i + 1 (cid:1) Γ (cid:0) m − ρ + i (cid:1) × i X r =0 ( − r (cid:18) ir (cid:19) Γ (cid:0) m − ρ + i + r + 1 (cid:1) Γ (cid:0) r − ρ − m − i + 1 (cid:1) ! (2.3) and ∞ X m =0 ∞ X n =0 ( − n ∆ m + n x m + n ( ρ ) m (2 − ρ − i ) n m ! n != ∞ X m =0 ∆ m x m ( ρ ) m m ! 2 ρ − m Γ (cid:0) (cid:1) Γ(2 − ρ − i )Γ (cid:0) m − ρ − i + 1 (cid:1) Γ (cid:0) m − ρ − i + (cid:1) × i X r =0 (cid:18) ir (cid:19) Γ (cid:0) m − ρ − i + r + 1 (cid:1) Γ (cid:0) r − ρ − m − i + 1 (cid:1) ! (2.4) ARJUN K. RATHIE
Derivations :
In order to establish the first result (2.1) asserted in thetheorem, we proceed as follows. Denoting the left hand side of (2.1) by S ,we have S = ∞ X m =0 ∞ X n =0 ( − n ∆ m + n x m + n ( ρ ) m ( ρ + i ) n m ! n !Replacing m by m − n and using the result[3, Equ.1, p.56] viz. ∞ X n =0 ∞ X k =0 A ( k, n ) = ∞ X n =0 n X k =0 A ( k, n − k )we have S = ∞ X m =0 m X n =0 ( − n ∆ m x m ( ρ ) m − n ( ρ + i ) n ( m − n )! n !Using elementary identities[3, p.58]( α ) m − n = ( − n ( α ) m (1 − α − m ) n and ( m − n )! = ( − n m !( − m ) n we have, after some algebra S = ∞ X m =0 ∆ m x m ( ρ ) m m ! m X n =0 ( − n ( − m ) n (1 − ρ − m ) n ( ρ + i ) n n !Summing up the inner series, we have S = ∞ X m =0 ∆ m x m ( ρ ) m m ! F (cid:20) − m, − ρ − mρ + i ; − (cid:21) We now observe that the series F can be evaluated with the help of theknown result (1.4) and we easily arrive at the right hand side of (2.1). Thiscompletes the proof of the first result (2.1) asserted in the theorem.In exactly the same manner, the results (2.2) to (2.4) can be established.So we prefer to omit the details.3. Corollaries
In this section, we shall mention some of the interesting known as well asnew results of our main findings.(a) In the result (2.1) or (2.2), if we take i = 0, we have ∞ X m =0 ∞ X n =0 ( − n ∆ m + n x m + n ( ρ ) m ( ρ ) n m ! n != ∞ X m =0 ∆ m ( − x ) m ( ρ ) m ( ρ ) m ( ρ + ) m m m ! (3.1) N A NEW CLASS OF SERIES IDENTITIES 5
This is a known result due to Srivastava[8]. Further setting ∆ m = 1 ( m ∈ N ), we at once get the result (1.1) due to Bailey.(b) In the result (2.3) or (2.4), if we take i = 0, we have ∞ X m =0 ∞ X n =0 ( − n ∆ m + n x m + n ( ρ ) m (2 − ρ ) n m ! n != ∞ X m =0 ∆ m ( − x ) m (cid:0) (cid:1) m (cid:0) ρ + (cid:1) m (cid:0) − ρ (cid:1) m m m !+ 2(1 − ρ ) xρ (2 − ρ ) ∞ X m =0 ∆ m ( − x ) m (cid:0) (cid:1) m (cid:0) ρ + 1 (cid:1) m (cid:0) − ρ (cid:1) m m m ! (3.2)which appears to be a new result.Further setting ∆ m = 1 ( m ∈ N ), we at once get another result (1.2) dueto Bailey.(c) In (2.1), if we take i = 0 , , · · · ,
9; we get known results recorded in [7].(d) In (2.2), if we take i = 0 , , · · · ,
9; we get known results recorded in [7].(e)In (2.1) to (2.4), if we get ∆ m = 1 ( m ∈ N ), we get known resultsobtained very recently by Rathie et al.[6].We conclude the paper by remarking that the details about the resultpresented in this paper together with a large number of special cases(knownand new), are given in [5]. References [1] W. N. Bailey,
Products of generalized hypergeometric series , Proc. London Math.Soc., 2 242–254, (1928).[2] W. N. Bailey,
Generalized hypergeometric series , Cambridge Tracts in Mathematicsand Mathematical Physics, Stechert-Hafner, Inc., New York, (1964).[3] E. D. Rainville,
Special functions , Reprint of 1960 first edition, Chelsea PublishingCo., Bronx, N.Y., (1971).[4] M.A. Rakha, A.K. Rathie,
Generalizations of classical summation theorems for theseries F and F with applications , Integral Transforms Spec. Funct. 22 (11), 823-840, (2011).[5] A. K. Rathie, On a new class of series identities, Submitted for publication, (2020).[6] A. K. Rathie, Y. S. Kim and R. B. Paris, On some new results involving productof generalized hypergeometric series with applications, Submitted for publication,(2020).[7] N. Shekhawat, J. Choi, A. K. Rathie and Om Prakash, On a new class of seriesidentities, Honam Mathematical J., 37(3), 339-352, (2015)[8] H. M. Srivastava, On the reducibility of Appell’s function F4, Canad. Math. Bull.,16 (2), 295-298, (1973).(Arjun K. Rathie) Department of Mathematics, Vedant College of Engineer-ing & Technology (Rajasthan Technical University), Village: Tulsi, Post:Jakhamund, Dist. Bundi, Rajasthan State, India,
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