Elementary Proofs for Convolution Identities of Abel and Hagen-Rothe
نویسنده
چکیده
By means of series–rearrangements and finite differences, elementary proofs are presented for the well–known convolution identities of Abel and Hagen–Rothe.
منابع مشابه
Multiplicate inverse forms of terminating hypergeometric series
The multiplicate form of Gould–Hsu’s inverse series relations enables to investigate the dual relations of the Chu–Vandermonde–Gauß’s, the Pfaff–Saalschütz’s summation theorems and the binomial convolution formula due to Hagen and Rothe. Several identitity and reciprocal relations are thus established for terminating hypergeometric series. By virtue of the duplicate inversions, we establish sev...
متن کاملABEL’S METHOD ON SUMMATION BY PARTS AND BALANCED q-SERIES IDENTITIES
The Abel method on summation by parts is reformulated to present new and elementary proofs of several classical identities of terminating balanced basic hypergeometric series. The examples strengthen our conviction that as traditional analytical instrument, the revised Abel method on summation by parts is indeed a very natural choice for working with basic hypergeometric series.
متن کاملMultinomial convolution polynomials
In 9] Knuth shows how to derive the convolution formulas of Ha-gen, Rothe and Abel from Vandermonde's convolution or binomial theorem for integer exponents. In the present paper, we shall rst present a short and elementary proof of the multi-extension of the above con-volution formulas, due to Raney and Mohanty. In the second part we shall present a multi-version of Knuth's approach to convolut...
متن کاملAbel–rothe Type Generalizations of Jacobi’s Triple Product Identity
Abstract. Using a simple classical method we derive bilateral series identities from terminating ones. In particular, we show how to deduce Ramanujan’s 1ψ1 summation from the q-Pfaff–Saalschütz summation. Further, we apply the same method to our previous q-Abel–Rothe summation to obtain, for the first time, Abel–Rothe type generalizations of Jacobi’s triple product identity. We also give some r...
متن کاملOn an extension of Riordan array and its application in the construction of convolution-type and Abel-type identities
Using the basic fact that any formal power series over the real or complex number field can always be expressed in terms of given polynomials {pn(t)}, where pn(t) is of degree n, we extend the ordinary Riordan array (resp. Riordan group) to a generalized Riordan array (resp. generalized Riordan group) associated with {pn(t)}. As new application of the latter, a rather general Vandermonde-type c...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Electr. J. Comb.
دوره 17 شماره
صفحات -
تاریخ انتشار 2010