Generating functions of Legendre polynomials Wadim Zudilin (based on joint work with Heng Huat Chan and James Wan) August 31, 2011 OPSFA 2011 Madrid Wadim Zudilin (CARMA, UoN) Generating functions of Legendre polynomials August 31, 2011 1 / 22
Overview Introduction 1 Main results 2 Applications 3 Wadim Zudilin (CARMA, UoN) Generating functions of Legendre polynomials August 31, 2011 2 / 22
Introduction Legendre polynomials Consider the Legendre polynomials P n ( x ), � n � � � − n , n + 1 1 − x � � x + 1 � − n , − n x − 1 � � � P n ( x ) = 2 F 1 = 2 F 1 � � 1 1 2 2 x + 1 � � n � 2 � x − 1 � m � x + 1 � n − m � n � = , m 2 2 m =0 where I use a standard notation for the hypergeometric series, ∞ � z n � a 1 , a 2 , . . . , a m � ( a 1 ) n ( a 2 ) n · · · ( a m ) n � � = n ! , m F m − 1 � z � b 2 , . . . , b m ( b 2 ) n · · · ( b m ) n n =0 and ( a ) n = Γ( a + n ) / Γ( a ) denotes the Pochhammer symbol (or rising factorial). Wadim Zudilin (CARMA, UoN) Generating functions of Legendre polynomials August 31, 2011 3 / 22
Introduction Brafman’s generating function The Legendre polynomials can be alternatively given by the generating function ∞ (1 − 2 xz + z 2 ) − 1 / 2 = � P n ( x ) z n , n =0 but there are many other generating functions for them. One particular family of examples is due to F. Brafman (1951). Theorem A The following generating series is valid: ∞ � � � � � � ( s ) n (1 − s ) n s , 1 − s 1 − ρ − z s , 1 − s 1 − ρ + z P n ( x ) z n = 2 F 1 � � � · 2 F 1 , � � n ! 2 1 1 2 2 � � n =0 where ρ = ρ ( x , z ) := (1 − 2 xz + z 2 ) 1 / 2 . Wadim Zudilin (CARMA, UoN) Generating functions of Legendre polynomials August 31, 2011 4 / 22
Introduction Bailey’s identity Theorem A in the form ∞ ( s ) n (1 − s ) n � X + Y − 2 XY � � ( Y − X ) n P n n ! 2 Y − X n =0 � s , 1 − s � � � s , 1 − s � � � � = 2 F 1 � X · 2 F 1 � Y � � 1 1 is derived by Brafman as a consequence of Bailey’s identity for a special case of Appell’s hypergeometric function of the fourth type, ∞ ( s ) m + k (1 − s ) m + k � m � � k � � X (1 − Y ) Y (1 − X ) m ! 2 k ! 2 m , k =0 � s , 1 − s � � � s , 1 − s � � � � = 2 F 1 � X · 2 F 1 � Y . � � 1 1 Wadim Zudilin (CARMA, UoN) Generating functions of Legendre polynomials August 31, 2011 5 / 22
Introduction Clausen’s identity Note that by specializing Y = X , one recovers a particular case of Clausen’s identity: � 1 � 2 � � 2 , s , 1 − s � � s , 1 − s � � � 4 X (1 − X ) = 2 F 1 . 3 F 2 � X � � 1 , 1 1 Wadim Zudilin (CARMA, UoN) Generating functions of Legendre polynomials August 31, 2011 6 / 22
Introduction Brafman–Srivastava theorem Theorem B (Brafman (1959), Srivastava (1975)) For a positive integer N, a ( generic ) sequence λ 0 , λ 1 , . . . and a complex number w, ∞ ∞ � k w z N 1 � x − z �� � � A n P n ( x ) z n , λ k P Nk = ρ N ρ ρ n =0 k =0 where ρ = (1 − 2 xz + z 2 ) 1 / 2 and � n ⌊ n / N ⌋ � � λ k w k . A n = A n ( w ) = Nk k =0 Brafman’s original results address the cases N = 1 , 2 and a sequence λ n given as a quotient of Pochhammer symbols (in modern terminology, λ n is called a hypergeometric term ). Wadim Zudilin (CARMA, UoN) Generating functions of Legendre polynomials August 31, 2011 7 / 22
Main results Ap´ ery-like sequences We extend Bailey’s identity and Brafman’s generating function to more general Ap´ ery-like sequences u 0 , u 1 , u 2 , . . . which satisfy the second order recurrence relation ( n + 1) 2 u n +1 = ( an 2 + an + b ) u n − cn 2 u n − 1 for n = 0 , 1 , 2 , . . . , u − 1 = 0 , u 0 = 1 , for a given data a , b and c . The hypergeometric term u n = ( s ) n (1 − s ) n / n ! 2 corresponds to a special degenerate case c = 0 and a = 1, b = s (1 − s ) in the recursion. n =0 u n X n for a sequence Note that the generating series F ( X ) = � ∞ satisfying the recurrence equation is a unique, analytic at the origin solution of the differential equation θ = θ X := X ∂ θ 2 − X ( a θ 2 + a θ + b )+ cX 2 ( θ +1) 2 � � F ( X ) = 0 , where ∂ X . Wadim Zudilin (CARMA, UoN) Generating functions of Legendre polynomials August 31, 2011 8 / 22
Main results Gist 1: Generalized Bailey’s identity Our first result concerns the generating function of u n . Theorem 1 For the solution u n of the recurrence equation above, define g ( X , Y ) = X (1 − aY + cY 2 ) . (1 − cXY ) 2 Then in a neighbourhood of X = Y = 0 , � ∞ �� ∞ n � 2 ∞ � � n 1 � � � � u n X n u n Y n g ( X , Y ) m g ( Y , X ) n − m . = u n 1 − cXY m n =0 n =0 n =0 m =0 Therefore, Bailey’s identity corresponds to the particular choice c = 0 in Theorem 1. Wadim Zudilin (CARMA, UoN) Generating functions of Legendre polynomials August 31, 2011 9 / 22
Main results Gist 2: Generalized Brafman’s identity Theorem 1 also generalizes Clausen-type formulae given recently by H. H. Chan, Y. Tanigawa, Y. Yang, and W. Z.; they arise as specialization Y = X . Following Brafman’s derivation of Theorem A we deduce the following generalized generating functions of Legendre polynomials. Theorem 2 For the solution u n of the recurrence equation above, the following identity is valid in a neighbourhood of X = Y = 0 : �� Y − X ∞ � n � ( X + Y )(1 + cXY ) − 2 aXY � u n P n ( Y − X )(1 − cXY ) 1 − cXY n =0 � ∞ �� ∞ � � � u n X n u n Y n = (1 − cXY ) . n =0 n =0 Wadim Zudilin (CARMA, UoN) Generating functions of Legendre polynomials August 31, 2011 10 / 22
Main results Gist 3: Special generating functions Theorem 3 The following identities are valid in a neighbourhood of X = Y = 0 : ( 1 � 2 n ∞ 2 ) 2 � (1 − X − Y )( X + Y − 2 XY ) � � X − Y � n · n ! 2 P 2 n ( Y − X )(1 − X − Y + 2 XY ) 1 − X − Y + 2 XY n =0 � 1 � 1 2 , 1 � � 2 , 1 � � � � = (1 − X − Y + 2 XY ) 2 F 1 2 � 4 X (1 − X ) · 2 F 1 2 � 4 Y (1 − Y ) , � � 1 1 � 3 n ∞ ( 1 3 ) n ( 2 3 ) n � ( X + Y )(1 − X − Y + 3 XY ) − 2 XY � � X − Y � P 3 n · n ! 2 � � ( Y − X ) p ( X , Y ) p ( X , Y ) n =0 � 1 � � − 9 X (1 − 3 X + 3 X 2 ) 3 , 2 � � p ( X , Y ) � = 3 (1 − 3 X )(1 − 3 Y ) 2 F 1 � 1 (1 − 3 X ) 3 � 1 3 , 2 � − 9 Y (1 − 3 Y + 3 Y 2 ) � � � × 2 F 1 3 , � 1 (1 − 3 Y ) 3 where p ( X , Y ) = (1 − X − Y + 3 XY ) 2 − 4 XY . Wadim Zudilin (CARMA, UoN) Generating functions of Legendre polynomials August 31, 2011 11 / 22
Main results Fred Brafman Fred Brafman was born on July 10, 1923 in Cincinnati, Ohio. He attended Lebanon High School (Ohio) from 1936 to 1940, then spent a year at Greenbrier Military School (Jr. College) before enrolling in the Engineering School at the University of Michigan in September 1941. He received a Bachelor of Science in Engineering (in Electrical Engineering) degree in 1943 and then a Bachelor of Science in Mathematics degree from Michigan in 1946. Brafman entered the graduate program in Mathematics in the fall of 1946 and compiled an outstanding academic record. He received an AM degree in 1947 and a PhD in February 1951 from the University of Michigan under the supervision of E. D. Rainville. After completion of his PhD, he was hired by the Wayne State University, by the Southern Illinois University, and then by the University of Oklahoma. Brafman had an invitation to visit the Institute for Advanced Studies (Princeton) which was not materialized because of his ultimate death on February 4, 1959 in Oklahoma. He solely authored ten mathematical papers, all about special (orthogonal) polynomials. Wadim Zudilin (CARMA, UoN) Generating functions of Legendre polynomials August 31, 2011 12 / 22
Applications Ramanujan’s series for 1 /π In 1914 S. Ramanujan recorded a list of 17 series for 1 /π , in particular, ∞ ( 1 4 ) n ( 1 2 ) n ( 3 (21460 n + 1123) · ( − 1) n 4 ) n 882 2 n +1 = 4 � π , n ! 3 n =0 ∞ ( 1 4 ) n ( 1 2 ) n ( 3 4 ) n 1 1 � (26390 n + 1103) · 99 4 n +2 = √ n ! 3 2 π 2 n =0 which produce rapidly converging (rational) approximations to π . Wadim Zudilin (CARMA, UoN) Generating functions of Legendre polynomials August 31, 2011 13 / 22
Applications Generalizations An example is the Chudnovskys’ famous formula which enabled them to hold the record for the calculation of π in 1989–94: ∞ ( 1 6 ) n ( 1 2 ) n ( 5 6 ) n ( − 1) n 3 � (545140134 n + 13591409) · 53360 3 n +2 = √ . n ! 3 2 π 10005 n =0 A more sophisticated example (which also shows that modularity rather than hypergeometrics is crucial) is T. Sato’s formula (2002) � √ √ √ ∞ � 12 n √ 5 − 1 = 20 3 + 9 15 � u n · (20 n + 10 − 3 5) 2 6 π n =0 of Ramanujan type, involving Ap´ ery’s numbers n � 2 � n + k � 2 � n � u n = ∈ Z , n = 0 , 1 , 2 , . . . , k k k =0 which satisfy the recursion ( n + 1) 3 u n +1 − (2 n + 1)(17 n 2 + 17 n + 5) u n + n 3 u n − 1 = 0 . Wadim Zudilin (CARMA, UoN) Generating functions of Legendre polynomials August 31, 2011 14 / 22
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